Chapter 5: Simultaneous Linear Equations
Reflection
You have learnt how to solve simultaneous linear equations by the graphical, substitution and elimination methods.
a) Compare these 3 methods in the aspects of accuracy, efficiency and convenience.
b) How do you decide which method to use to solve a system of simultaneous linear equations if the method is not specified? Give examples to explain your decisions.
Your task:
- This assignment is graded (upon 10 marks)
- Marks will be awarded on 1) Content & Knowledge, 2) Effort & participation
- Click on the 'comments' link below to do your assignment. Remember to write in your register number as well as your name!
45 comments:
Hi all,
Like the last time, after you leave your comments, they will not appear yet because I need to approve them first.
I will release all the entries after I have received all 41.
(ie on Saturday, 14 June)
So from 15 June onwards you will be able to view all entries.
Pls take this seriously, as you all already know by now, the marks will be counted in the CA and it can pull your overall marks up/down rather significantly!
Ok I have done away with the approving step, after some feedback from your classmates. So pls don't copy, do this assignment with integrity, I'm sure all of you have your own opinions on this topic.
There isn't really an absolute correct / wrong answer, so don't worry & gd luck!
Wang Shu Xian(30) of class 2A here. the journal of the 'which method u usually like to use' thing.
I prefer to use Elimination Method instead of using the graphical or substitution method as generally speaking, the elimination method is more accurate and more efficient than the other methods.
Graphical method: as you said in class, is not very accurate because of human error like the line or marked point may be off and thus, resulting in an inaccurate answer. Also, Graphical method cannot measure answers to decimal points like the two other methods as graphs has its limit, having only ten 'small lines or box' in 2 'big box' on the graph paper. Another thing, Graphical method is not very efficient as you have to form your own equation, find your own range of where the X and Y axis should be at, and sometimes when you guessed the range incorrectly, you will have to redo the points which result in wasting time to do the same question twice or more times.
Substitution Method: Although it is accurate in solving the equation if there are no calculation mistake, it is not as efficient as the elimination method because of the need to form a basic of two equations, then further 'modify' one of the equations to form a new equation to substitute inside one of the equations. If you are not careful, it is easy to make mistakes when you change one equation to substitute into another as you have to 'play' with the signs and algebra. Sometimes, it gets confusing and you may change the equation wrongly, resulting in an incorrect answer. The time taken to change the equation, substitute it into another equation, simplify the equation and finally solve it takes up a lot of time thus, it is not very efficient.
Elimination method: Although like the other two method, you have to form your own equation. However, it is more accurate than the graphical method and is more efficient than the sub. method. For elimination method, you just have to form 2 equations and from there, multiply or divide one of the equation so that one of the co-efficient (or whatever term you use for the number infront of the algebra) of the same algebra in the two equations are the same. from there on, you juz have to add or subtract one of the equation from another (depending on the algebra being positive or negative). Then, you can find one of the algebra. After that, you just have to sub. the number into the equation to find out the other algebra (like what you do in the substitution method).
Therefore, the elimination method is more convenient, accurate and effficient than the two other methods.
**sorry for typing error or bad use of english**
Zeng Fan Jun(41)
Maths Journal
Theory:
Simultaneous equations are two or more equations that have the same variables and the same solutions.
I shall describe 3 methods to solving them.
We will first talk about the elimination method.
Elimination is basically a few steps to follow, simple for beginners to understand:
1) Multiply one of the equations by a constant in order to make the coefficients match. (There are a lot of choices to choose from)
2)Add the equations or subtract the equations to get the answer of a coefficient.
3)Substitute it into one of the equations. And you're done.
Substitution is a more complex method, therefore, more difficult to understand.
1) First, isolate one of the variables on one side of the equation.
2) Substitute for the isolated variable in the other equation.
3) Solve the other equation for the other variable.
4) Substitute the known value of the other variable into the first equation derived in step 1.
Lastly, the graphical method, which is not recommended to use.This method has low accuracy as any mistake in drawing the graph causes a wrong answer.
In this method,
1)Plot the simultaneous equations given.
2)Read off the point at which they cross.
3)Interpret your answers from the graph.
My reflections on this chapter:
Along with many other students, i prefer the elimination method. It is the simplest method. Commonly, students first multiply the equations. Once they get the same coefficients, they just have to do some simple subtraction or addition to solve one of the variables. Substitute to solve the other.
However, in substitution, the chances of messing up are higher if you are not careful.When you solve one of the variables, you have to substitute it into the other equation and multiply. The trap that the teacher sets for you in exams lie here. A calculator is unable to calculate an equation multiplied by a variable. Thus, you have to calculate it yourself. You will only be able to get the correct answer if you multiply everything correctly and get the first equation of the variable correct. Therefore, this method is more risky. It requires a higher level of thinking. It takes longer time. The disadvantages of this method decreased its popularity among students.
The graphical method is a method that is useful. However, the drawing of graphs are time consuming. There is a chance that time pressure will cause incorrect and inaccurate plotting of graphs.
This is jeannie siong (24)
I will put my name as xiongsiong the next time so please take note
I would choose the elimination method, as it is the easiest and most accurate method to use
For the graphical method, it is not accurate, as it does not show to the decimal place. though it usually accepts a higher range of numbers, for eg. 0.9-1.1, it is really difficult to decide which is the number u have exactly in yr graph as the 'big box' and 'small boxes' can have huge differences and also from you and yr friends. Sometimes, the line of the graph might even cut in between the small boxes, when it comes to that, u might have yr answer in difference of 0.1. For example, back to my answer above, 0.9-1.1, due to my horrible drawing of my graph, i got my answer as 1.2 and therefore, got my mark minus away. also, when using the graphical method, it is the one with the least convenience! due the fact that you cannot get a graph wrong as it is quite obvious when you draw it wrongly, when you see that its is wrongly drawn, you have to rub it away resulting in a waste of time, which we do not have especially during exams which causes people to have not enough time to do the remaining questions. Therefore, i think i will not ever use the graphical method if possible.
Next, for the substitution method, I do not really like to use it as it might result in fractions, which you might easily make careless mistakes. For examples, using the substitution method, y= 2x-4x2+9x3/x. (I know that the equation is very hard to understand, but you don’t really have to, you just have to get the point) which is like very confusing. AND have the huge tendency of making a very major mistake causing the whole answer and working to be wrong. Also, it will be very difficult to check for errors too as also, due to the fact that the equation becomes very confusing. Due to all the confusing fractions cause by making an unknown the subject, you might also reach really weird and inaccurate numbers which you have to recheck all your workings again, again causing a waste of time which you do not have during your tests and exams. All in all, for extremely careless people like me, I would not advise you to use the substitution method as it would be the most major mistake you might ever make during your exams.
Lastly, it is the elimination method. You simply have to plus, minus, times and divide, which usually do not include big numbers! You would usually meet numbers like 2,3 ,4 and so on, which is usually impossible to reach 10+. (You get my point)It is the simplest method as you do not have to include fractions! Even if you have an equation with loads of fractions, you can simply remove it by multiplying the HCF, which is simply IMPOSSIBLE to get a careless mistake in it, unless you really are THAT careless, because the denominators usually will not be extremely big numbers and is manageable. Next, the only thing you need to do is spot out for the same x or y and eliminate it! The easiest thing is that, the questions, for problem sums, are usually organised in a way that it is very easy to find the two equations needed to use the elimination method! So, I will advise all people to use the elimination method after reading my comment.
As the elimination method is the most convenient, accurate and efficient method, I would definitely use this method during my examinations and it has never ever failed me. Thank You.
Tan Li Hui (25)
In simultaneous linear equations, there are three methods used to solve a pair of simultaneous equation. They are by the graphical method, substitution method and by the elimination method.
In my opinion, I think that the elimination method is the easiest, most accurate and convenient method to use.
Compared to the graphical method, the elimination method is a lot faster while considering the time factor. In the graphical method, you would have to use a pencil, ruler, eraser and a calculator to solve the question. But as for the elimination method, you would only need your pen and calculator. Using the graphical method, you would have to draw a table, then use the equations given to you in the question, sometimes you have to even form an equation yourself. After that, if you do not calculate the answer correctly, the graph that you drew would be wasted and you would have to re-count and redraw everything again. This would waste a lot of time while you are doing your maths paper! But as for the elimination method, you would not have to deal with any meddlesome graphs; you would just have to use your pen and calculator. Even if you make a mistake, you would only have to recheck and make amendments. Also, the graphical method is sometimes not accurate because if the answer you get is beyond one decimal place, e.g., 2.23, you cannot really divide the small points in your graph into 10 small parts but as for the elimination method, you need not make any graph, thus even if you get an answer with a decimal place, you can just leave the answer as it is.
Comparing the elimination method to the substitution method, I think that the elimination method is easier as it uses shorter steps. By the elimination method, after you eliminate either x or y, you can immediately find out the answer of what you eliminated. This way, you would not have to waste time substituting one equation into another, which the substitution method does. Other than that, I think that substitution method would seem to be more difficult to solve as you are substituting two equations together. After that, you have to solve the rest of the equation, adding and subtracting here and there. As compared to the elimination method, you just have to either add or subtract two equations together. The result you get would only have either x or y. Although you would have to be extra careful while adding or subtracting the two equations, the elimination is actually a very straightforward method. After you find out what x or y is, you would just have to substitute it into another equation.
Thus, the elimination method is the most accurate, efficient and convenient method to use.
Jennifer Gwee(7)
okay.i totally dont understand why people like the elimination method so much-.-
There are 3 types of method to solve simultaneous equations, namely, graphical method, substitution method and elimaination method.
GRAPHICAL METHOD
If a solution exist in the equations, it will appear as the coordinates of the graphs of the simultaneous equations.But the MAIN drawback of the graphical method is that we may not get the EXACT solution of the equations.
For example: Solve the simultaneous equations
6x+9y=1
and x+y=1
graphically
6x+9y=1
sol:X -2 0 2
Y 1.44 0.11 -1.22
x+y=1
sol:X -3 0 3
Y 4 1 -2
The coordinates of the point at intersection of the 2 graphs are not integers. Hence, we can only read the APPROXIMATE values of the coordinates which are corrected to one decimal place, and time is needed to draw the points and plot the graph.
SUBSTITUTION METHOD
Exact solution can be obtained via this method and I personally think that substitution method is the easiest way to solve simultaneous equations as there is no need to draw graphs or eliminate equations which might lead to unwanted careless mistakes. Substitution method just involves substitution of an equation into another to solve.
Eg:
y=2x+5-----(1)
4x-2y=10---(2)
sub(1) into (2):
4x-2(2x-5)=10
4x-4x+10=10
10=10
(eg of solution with infinite solutions)
ELIMAINATION METHOD
It solves the simultaneou
equations by eliminating a particular unknown by adding or substracting the equations. The way to solve equations by using this method is to:
1)Identify coefficients with the same numerical value in the two equations.
2)Add/Subtract the equations to eliminate the first unknown
3)Solve for the second unknown
4)Substitute the value of the second unknown to solve the first unknown
eg: 5x-9y=32----(1)
3x-8y=27----(2)
(1)x3:15x-27y=96-----(3)
(2)x5:15x-40y=135----(4)
(4)-(3):-13y=39
y=-3
sub y=-3 into (1):
5x-9(-3)=32
5x+27=32
5x=5
x=1
So, x=1, y=-3
As you can see, the elimination method is longer and mistakes might appear if you forget to multiply the WHOLE equation and more time is needed to solve the equations as compared to the substitution method.
To solve a system of linear equations, we must first find out the 2 unknowns, then form 2 linear equations and then start solving. For me, I'll use the substitution method as it is faster, easier and not that easy to make careless mistakes. But if the need arises, I'll opt for elimination method.
eg:
A two-digit number is equal to 4 times the sum of its digits. If the digits of the number are reversed, the new number formed is 27 more than the original number. Find the original number.
SOLUTION:
Let x be the tens digit
Let y be the units digit
original no:10x+y
sum of digits:x+y
no formed by reversing digits:10y+x
From the given info;
10x+y=4(x+y)------(1)
(10y+x)-(10x+y)=27----------(2)
from 1:10x+y=4x+4y
6x-3y=0
2x-y=0---------------(3)
from 2:9y-9x=27
y-x=3--------------(4)
x=3
sub x=3 into (4);
y-3=3
y=6
original no:10(3)+6
=36
there.DONE.finally-.-
There are basically three ways of solving linear equations .Graphical method , substitution method and elimination method.
Graphical Method
Many of us would not choose this method as the solution is not obvious.
The solution to the simultaneous equations is their point of intersection.
The graph of a linear equation ax+by = c is a straight line.
Two distinct lines always intersect at exactly one point unless they are parallel (and that is = having the same slope)
The coordinates of the intersection point of the lines is the solution to the simultaneous linear equations describing the lines. So we would normally expect a pair of simultaneous equations to have just one solution.
So, if the equation is true, the lines are parallel, they do not intersect, and the system of linear equations has no solution.
Elimination
This method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable.
Okay ,
for example:
Solve the following pair of simultaneous linear equations:
Equation 1: 2x + 3y = 8
Equation 2: 3x + 2y = 7
Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. An easy choice is to multiply Equation 1 by 3, the coefficient of x in Equation 2, and multiply Equation 2 by 2, the x coefficient in Equation 1:
3 * (Eqn 1) -> 3* (2x+3y=8) -> 6x + 9y = 24
2 * (Eqn 2)-> 2 * (3x + 2y = 7)-> 6x + 4y = 14
AS YOU CAN SEE,
both equations now have the same leading coefficient = 6
Step 2: Subtract the second equation from the first.
6x + 9y = 24-(6x + 4y = 14) =
5y = 10
Solve this new equation for y.
y = 10/5 = 2
Sub y=2 into Equation 1.
2x + 3(2) = 8
2x + 6 = 8
2x = 2
x = 1
Solution: x = 1, y = 2
ISNT THAT MUCH SIMPLER THAN THE GRAPHICAL METHOD? For graphical method , we'll get confused easily .
Now, moving on to the
The Substitution Method
This method involves making one variable the subject of an equation, and substituting it into the other equation to obtain an equation with a single variable.
Equation 1:-3x+y=5
Equation 2:4x+y=-5
Write y as the subject in Eqn 2
y = ( -5-4x )
Sub y into (1) to find x
-3x+1(-5-4x)=5
SOlve x.
x=-1.43(2 decimal place)
Solve y.
y=0.71(2 decimal place)
Here it was easy to first make y the subject of Equation 2, as the coefficient of y is 1. We could still solve the problem in this way even if the coefficient of y was not 1, it would just involve more algebra .If Equation 1 had a y coefficient of 1, it would be best to make y the subject of equation 1 and then substitute into Equation 2. Similarly, if x had a coefficient of 1 in either equation, you could rearrange that equation to make x the subject of the equation and the substitute for x in the other equation.
However the disadvantage of this method is that, there might be transfer error, well, for careless people like me. THAT'S WHY,I THINK THAT ELIMINATION IS THE EASIEST METHOD OF ALL :D
Firstly I shall start by defining each method.
The substitution method requires us to choose 1 unknown from 1 equation and make it the subject of the equation. Following, we will have to substitute this expression into the other equation given. Solve the equation and we will find the 2nd unknown. Substitute the solution frm the 2nd unknown and we will be able to find the 1st unknown.
1 example:
2x+y=25----(1)
3x-4y=-1---(2)
Frm (1),
y=25-2x----(3)
Sub.(3) into (2),
3x-4(25-2x)=-1
3x-100+8x=.1
11x=99
x=9
Sub.x=9 into(3)
y=25-2(9)
=7
The solution is x=9 and y=7.
In my opinion, the substitution method is faster but it has a higher chance of making careless mistakes throughout if we are not careful enough .Also, sometimes when the unknown that we are making as the subject is in fraction form which makes it difficult for some students to solve as they can make careless mistakes throughout. However, we are able to obtain 1 of the unknown by just substituting the other ans into the 3rd equation quickly. This makes the job simpler for the 2nd half part.
Now let’s look at the elimination method:
Like the name goes, we have to eliminate 1 of the unknown first. Firstly, we need to compare the coefficients of the unknowns. Adjust the coefficients of the 2 equations to be the same by multiplying them. Then, eliminate 1 of the unknowns by adding or subtracting. Solve the resulting equation and solve for the unknown. Substitute the value into any of the original equation to obtain the other unknown.
1 example:
3x+y=8----(1)
2x+3y=17—(2)
(1)x 3: 9x+3y=24---(3)
(3)-(2):7x=7
x=1
Sub. x=1 into (1),
3(1)+y=8
3+y=8
y=5
The solution is x=1 and y=5
In my opinion, the elimination method is more efficient as we can eliminate one of the unknown easily by just making 1 of the coefficients of the unknown the same and add it or subtract it. This makes the job simpler for us to find the 2nd unknown unlike the substitution method where we may make careless mistakes when making one of the unknown the subject. This method also allows us to obtain the 2nd unknown by substituting the unknown found into any of the equations which we are comfortable with.
However, using the elimination method is long-winded at times we may get many equations and makes it confusing
Lastly, let’s look at the graphical method.
Firstly, we have to plot tables for the 2 different equations given. After plotting the tables we will have to plot the graph. The coordinates where the graphs intersect will be the solution of the simultaneous equations.
In my opinion, the graphical method is both inconvenient and inefficient. If there is a point where the results we get from the table is a fraction or recurring decimal it will be difficult for us to plot the graph in the paper and as a result we may get an inaccurate answer. Also, when we realize that we have made a mistake amid plotting the graph, we will have to redo everything and erase the graphs, which is tedious and a waste of time, Thus the graphical method is not advisable. However, we are also able to drive frm the graph the solution at times where the ans is obvious eg. (-1,1).
From all of the above, I prefer to use the substitution method. It is more faster and easier to obtain the 2nd unknown as we have already work out the 3rd equation and get the 1st unknown. However, at times where the coefficients are the same for the unknown or when a complex fraction occur when we are making 1 unknown the subject during the substitution method, I would use the elimination method. Generally, it depends on the situation which would allow me to choose the method that I prefer. :)
Eliza Ng Sue Ying(3) 2A
a) First for the graphical method,
Ms Chum said that graphical method may not be that accurate as sometimes the graphs are drawn wrongly and the intersection points might thus be different. Therefore, it might not be that accurate. Graphs also can't give a very exact number of the intersection point as it can only measure up to one decimal point. Sometimes the intersection points have more than one decimal place. As this method needs time to draw the graph and draw the graph table, it would normally take a longer time. Therefore, it is no as efficient as the other two methods. As it would need graph paper for drawing, it would not be as convenient as the other two methods which only needs the same paper but using methods.
Secondly for the substitution method,
the substitution method is quite accurate as it requires working and any amount of decimal place can be calculated. Unless a person has calculation mistake, if not the answers will always be correct. As it would probably take alot of steps to substitute the equations, it would take a longer time to work out compared to the elimination method. It is quite convenience as it does not require any other things other then the working paper a pen, correction tape and a calculator.
Lastly for the elimination method,
it as almost the same as the substitution method. It has the same accuracy unless the person made a careless mistake. It is more efficient than graphical method and substitution method as it requires less working and no drawing is required at all. It is as convenience as the substitution method as it does not really require anything else except for a pen, correction tape and a calculator for calculaton.
b) For me, i would usually choose to use the elimination method if the method is not specified. elimination method is quite accurate, efficient and is rather convenient compared to the other two methods. However, my choose would vary when there are different questions. Some questions can be done easier by using the substitution method rather than the elimination method. However question like,
x + y = 156 ------- (1)
2y - x = 57 ------- (2)
(2): -x + 2y = 57
(3) + (1):
3y = 213
y = 71
Sub y = 71 into (1):
x + 71 = 156
x = 156 - 71
x = 85
Therefore x = 85, y = 71.
Hence, the choice of method would vary from different questions. But the most common used is elimination method.
there are 3 types of methods to solve simultaneous equations : graphical method, substitution method and elimination method.
graphical method is not as efficient as the other two methods. a long time is needed to draw the graph and plot accordingly. if the graph is plotted wrongly, you will need to redraw and calculate again, which takes up much effort and time.
unlike the graphical method, substition method is simply forming two equations, one which is either x or y and the other is the one being substituted. then solve directly. it is simple but sometimes it can be difficult to decide which to be x or y and which to be substitued.
elimination is the best method in my opinion as it is just multiplying the numbers to get the two coefficients the same and subtract them away. this method is slightly faster and definitely more efficient than graphical method. if you are careful, it is unlikely to make careless mistakes, so the elimination method is the best method of the three!
Hi!Xin Yi(13) here!I am going to discuss about solving simultaneous equations.
Firstly,I'm going to compare about the three methods of solving simultaneous equations.
For graphical method,it is efficient as when you plot the graph, you can roughly know what is the number of x or y.However, it is not accurate as everybody have different scale and sometimes if you never plot the graph properly,you may find the wrong answer.It is also not convenient as you need to use a ruler to draw the graph and you have to be very neat to join the plottings together in order to find the correct answer.You also need a long time to draw the graph.
For substitution method,it is accurate as you are using workings to solve the sums unless you make any careless mistakes.It is efficient and convenient as all you need to do is just to substitude the equation into x or y and make only one equation with only one alphabet.
For elimination method,it is accurate as you are using workings to solve the sums unless you make any careless mistakes.It is not efficient and inconvenient as you need to use lowest common multiple to do the sums and you need to do a lot of equations sometimes.
I would use substitution method to solve a system of simultaneous linear equations if the method is not specified as it is accurate,efficient and convenient.
There are three types of methods to solve for simultaneous equations. By graphical method, substitution method and elimination method.
Substitution method-
In this method, we solve one of the equations for one unknown and substitute this expression into the other equation to solve for the remaining unknown.
Graphical method-
In this method, we draw the graphs of both equations on the same axes and we read off the values of the variables at the point of intersection of the two lines (the coordinates of the point of intersection gives the solution to the simultaneous equation)
Comparing graphical method with substitution method:
Graphical method requires a graph to be drawn in order to get the solution. Risks are involved as one might make careless mistakes while plotting the graphs and sometimes have to estimate where the points should lie when the number is not a whole number. (Eg. 0.14, 2/3) thus the answer is not an accurate one. Substitution method is a better method as we just need to substitute a particular number to an equation to get the answer.
Elimination method-
In this method, we first eliminate one of the 2 unknowns. Either one of the unknowns can be eliminate first. Solve the resulting equation in one unknown. Find the value of the unknown. Substitute the value obtained into any of the original equations to solve for the other unknown.
Comparing substitution method with elimination method:
When we substitute a number into an equation, we often will make careless mistakes when the number is not accurate or is not a whole number. (eg. 0.14or 2/3) All the substituting will make us easily confused thus not calculating out the correct answer. Elimination is the best method among the three methods as the numbers are all multiply by the common factors and they are often whole numbers that are less confusing. The numbers are easily calculated thus the risk of making careless mistakes is being lowered down.
Take this as an example to illustrate what I mean:
Solve this simultaneous equation.
By elimination method:
2x-3y=9------------- (1)
3x+2y=7----------- (2)
(1) X3: 6x-9y=27------------ (3)
(2) X2: 6x+4y=14---------- (4)
(4)-(3):
13y=-13
y=-1
Sub. y=-1 into (1):
2x-3(-1) =9
2x+3=9
x=3
The required solution is x=3, y=-1
By substitution method:
2x-3y=9------------- (1)
3x+2y=7----------- (2)
From (1):
2x=9+3y
x=9+3y/2--------- (3)
Sub. (3) into (2):
3(9+3y/2) +2y=7
27+9y/2+2y=7
27+13y=14
13y=14-27
y=-1
Sub. y=-1 into (2)
3x-2=7
3x=9
X=3
The required solution is x=3, y=-1.
From the example, we can see that solving the question by elimination method is more efficient than substitution method. Less steps are involved and hence careless mistakes could be omitted.
I agree that solving simultaneous equation by the three methods will arrive at the same answer. But for me, I would definitely choose to use elimination to solve simultaneous equation unless the question stated otherwise. It is the easiest way as we only need to multiply the equations by its common factors instead of tackling with the fractions often present in substitution method.
Done By: Irene Tan Mei Chin (26)
i am cheng york (34)
ELIMINATION METHOD
1. in this method, we first eliminate one of the two unknown. Either one of the unknowns can be eliminated first. pick the
one that is easier.
2.(A) Compare the coefficients of the unknown we want to eliminiate. If they have the same coefficients, then eliminiate by
adding or subtracting,
(B) If the coefficients of the unknown to be eliminated are different:
(i) adjust one of the equation by mutiplying with an appropiate number or
(ii)adjust both equation by mutiplying with a different number for each equations.
(iii)Eliminate one of the unknowns by adding or subtracting,
(C)Solve the resulting linear equation in one unknown. Find the value of this unknown.
(D)Substitute the value obtained into any of the original eqution to solve for the other unknown.
(E) Check that the values you have found for both unknowns satisfy the original equation that you have not used. YOu can
check mentally or writing down on paper.
NOTE : If you have to solve a pair of simutaneous equqtions by elimination, the unknowns must be on the same side of the
equation and in same order. IF not rearrange the equation before you start.
E.g. Rearrange 2x=3y+10 to 2x-3y=10
2y+3x=10 3x+2y=28
from this methods there are likely to hav no problem, and have most 100% accurancy, if there is no careless mistakes.
SUBSTITUTION METHOD
1 in this method, we solve one of the equation for one unknown ans substitute this expression into the other equation to
solve
for the remaining unknown.
2 steps to solve simultaneous equations by substituion:
(A)Choose one of the equations and make one of the unknowns the subject of the equation.
(B)substituite this expression into the other equation to obtain an equation with only one unknown. Solve the equation to
find
this unknown.
(C)Substitute the solution into the expression from part (a) to find the second unknown.
(D)check that the values obtained satisfy the other original equation that you have not used.Again you can do a quick
mental calculation or write it down on paper.
But it may confused the wirter by using 2 variables at a same time.
GRAPHICAL METHOD
consider two linear equations :
x+y = 6
x-2y=3
every point on the graph of each eqation is a solution of the equation. hence each of the above equations has an infinite
number of solutions. However, do they have any common solutions between them?it depends.
It has a pair of values of x and y that satisfy both equations. Hence, if a solution exist, it is th e coordinates of the
point of intersection of the graphs of the simultaneous equations.
If a solution exist in the equations, it will appear as the coordinates of the graphs of the simultaneous equations.But the
MAIN drawback of the graphical method is that we may not get the EXACT solution of the equations. As usually we draw, and tend to get
incorrect answers, due to human errors and other problems.It would bee wrongly drawn and in a result of giving inaccurate
answer.Graph also can't be 100% accurate as it is given up to 1 decimal point.it would take us a ot of time and is very
time-consuming.therefore i do not recommend student to use graphical methods.
I would like to use elimination method as i think it is the most conveinence, accurrate, and deffience.
please seek forgiveness for punctuation and grammar errors.
There are several ways to solve linear equations, namely the graphical, substitution and elimination method. In my opinion, the best method to solve a linear equation would be the one that requires the least amount of work. However, that will depend on the particular equation that you’re trying to solve.
Graphical Method:
The graphical method is usually the hardest, since it takes up rather a lot of time and effort. However, it may be the easiest method to some, considering the fact that it requires less mathematical skills than the other two methods but there is always room for error when trying to read a point of intersection from a graph. This method is not as accurate as the other methods when it comes measuring answers to a decimal point. So using the graphical method is handy when you just need an approximate answer, but if you need a lot of precision it is better to use the other two methods.
Substitution Method:
This method is rather simple because all you need is a pen and a calculator. Although the steps to solving the equation using this method are simple, it can be rather tedious due to the multiple steps required. Also, there is a higher chance of making careless mistakes while doing the multiple steps.
Elimination Method:
This is most simplified method and like the substitution method, all you need is a pen and a calculator. There fewer steps because you just need to multiply the whole equation.
Substitution is a convenient method to solve systems of equations when one of the two equations has a coefficient of 1 on either x or y. If no variable has a coefficient of 1, substitution becomes a bit more cumbersome. Consider the following example,
3x − 4y = 7,
−2x + 5y = 0.
Since neither equation has a coefficient of 1 on x or y, we would have to solve for x or y in order to make a substitution. For example, we could solve for x in the first equation as,
This is an awkward looking expression for x that we would have to substitute into equation 2 as,
While we could go on to solve for y, you can see that substitution has produced a messier expression, and it is easy to make an error with such an expression. Hence it would be easier to use elimination.
In this example, we will eliminate the variable x. The coefficient of x is 3 in equation 1 and 2 in equation 2. Therefore, if we wish to eliminate the variable x, we need to make the coefficients of x equal to one another and subtract the equations or opposite of one another and add the equations. The easiest ways to do this is to multiply both sides of the first equation by 2 and both sides of the second equation by 3 as,
2 (3x − 4y) = 2 (7) ,
3 ( −2x + 5y) = 3 (0) ,
which gives,
6x − 8y = 14,
−6x + 15y = 0.
Now the coefficient of x in equation 1is 6, and the coefficient of x in equation 2 is -6. By adding the two equations together, we can eliminate x altogether as shown,
−8y + 15y = 14 + 0.
Therefore, we find that
7y = 14
y = 2.
Now we need to find the x-coordinate of the point of intersection. We can do this by substituting the value of y, in this case y = 2, into either linear equation as follows,
Using the other linear equation we arrive at the same result,
Thus, we find the solution to this system of linear equations to be (x, y) = (5, 2).
Ng Yee Hang(37)
Of all methods, i of course will prefer the elimination method like other students.
First of all, The graphical method is quite tedious.You would need to draw a graph which most students would complain since drawing a graph will need accuracy and if you draw it wrong you will need to redo it again.In normal assessments, re-drawing is ok.BUt in exams you would not have the time to re-draw.IF you drew wrongly you will not get an accurate answer. Also a calculation mistake in x and y will cause u to get an inaccurate answer.For example, x=2y-1
so if the question states that y = 2, the answer should be x=2(2)-1
=3
But if u state the answer as 5, the graph drawn will be incorrect.So in conclusion, the graphical method is not efficient and accurate at all.Also it is quite tedious and if the paper color is too bright you would have a hard time planning the points on the paper therefore i would not choose the graphical method.
Now for the substitution method,
it requires you to find what x or ystands for, before substituting x or y into any of the equations.A minimum of 3 equations is needed.
For example, x+3y=8----(1)
2x+5y=13----(2)
In this case, we should find what x stands for as it is easier so,
From (1), x=8-3y----(3)
Now that we have found what x stands for we can substitute (3) into (2).
Sub (3) into (2),
2x+5y=13
2(8-3y)+5y=13
16-6y+5y=13
16-y=13
y=3
We have found the value of y next we proceed to sub y=3 into any of the equations.
Sub y=3 into (1)
x=8-3y
x=8-3(3)
x=-1
Now we have found the value of x and y.Compared to the graphical method earlier on, the substitution method allows you to find the exact value of x and y but it requires you to express x in terms of y or y in terms of x.After that, you then find the exact value of y or x.For the last step, after you have found y or x, you then proceed to find x or y depending on whether you expressed x in terms of y or y in terms of x.So the substitution method is actually more efficient and more accurate than the graphical method.
Last but not least, the elimination method.Unlike the substitution method, it does not require you to express x in terms of y or y in terms of x.For example,
5x+2y=18----(1)
6x-2y=26----(2)
Since the coefficients of y in these equations are +2 and -2 respectively, we add equations (1) and (2) to eliminate the terms in y.
Adding (1) and (2),
(5x+2y)+(6x-2y)=18+26
11x=44
x=4
Substituting x=4 into (1)
5x+2y=18
5(4)+2y=18
20+2y=18
2y=-2
y=-1
As you can see, the elimination method does not require expressing x in terms of y or y in terms of x at all.So in a way, it saves more time than expressing x in terms of y or y in terms of x.So it is more efficient than the substitution method or graphical method.It is also accurate than the substitution method since it does not require the expressing of terms,it is actually does not allow the one taking the examination to slip up hence in conclusion i prefer the elimination method in terms of accuracy, efficiency and convenience.Thank you for your time and hope that my reflection is useful
I personaly prefer the elimination method as i feel that it is the most efficient and accurate method.
GRAPHICAL METHOD
To find the solution of simultaneous linear equations using the graphical method, we first plot the equations on the graph, then we identify the point whre the two lines cross and we get the solution.
This method is not as accurate and efficient compared to the substitution method and the elimination method.
This is because we may get the wrong answer easily if we do not draw the lines accurately thus affecting the answer we get and causing marks loss. Moreover, we need to find the range used to draw the graph if it is not given. If we use the wrong range, the two lines may naver meet, thus causing the need to change the range and re-calculating. Finding the right range and drawing the graph are both very time consuming.
It is not really accurate as the answer we get is unable to have decimal points and we have to find the answer base on what we see, thus, parallax error may occur. This method is thus only useful when we need an approximate answer.
SUBSTITUTION METHOD
In this method, algebra is used. We first substitute one variable into another variable before solving the equations.
Substitution is a more convenient method to solve systems of equations when one of the two equations has a coefficient of 1 on one of the unknown.
For example: x+3y=8---(1)
2x+5y=13---(2)
First, we change the x into y.
From (1), x=8-3y---(3)
Substitute x=8-3y into (2),
2(8-3y)+5y=13
16-6y+5y=13
16-y=13
y=3
Substitute y=3 into (3),
x=8-3(3)
x=8-9
x=-1
therefore the solution is x=-1 and y=3
ELIMINATION METHOD
Algebra is also used in this method. Elimination method is eliminating a particular unknown by adding or subtracting the equations.
For example: x+3y=8---(1)
2x+5y=13---(2)
(1)x2: 2x+6y=16---(3)
(3)-(2): (2x+6y)-(2x+5y)=16-13
y=3
From (1), x=8-3y---(4)
substitute y=3 into (4),
x=8-3(3)
x=8-9
x=-1
therefore the solution is x=-1 and y=3
Although I prefer the elimination method, there are times when I choose the substitution method over the elimination method even when the method used is not specified. These are times when one of the equations has a coefficient of 1 on one of the unknown. This is because the time taken to calculate the solution using the substitution method is less than that of using the elimination method as the steps used are less. This is based on the examples I used for the substitution method and the elimination method. Thus, I feel that different methods need to be used based on different situation in order to achieve better results.
Sim Jia Lin(22)
2A (:
I prefer using the elimination method instead of using graphical method and substitution method as i think that it is a much easier way of getting the answer and you won't go wrong if you follow the sequence carefully. It is more efficient and more reliable and accurate. I understand elimination method better then the 2 other methods, so i will definitely choose elimination method to solve a system of simultaneous linear equations if the method is not specified. However, there are some cases where i will use the substitution method. This is all due to the questions and equations stated in the questions.
GRAPHICAL METHOD:
To find the solution of simultaneous linear equations using the graphical method, we need to first plot the equations on the graph, next we find the intersections of the two lines.
i think that this method is not as accurate and efficient as compared to the substitution method and the elimination method. We may get the wrong answer easily if we do not draw the lines carefully and accurately. The lines must be drawn accurately at the points and sometimes it is real hard and due to human error, we may not draw the lines correctly. This can result in loss of marks. Adding on,we will need to find the range used to draw the graph if it is not given. It is very troublesome and if we are careless and we use the wrong range, the two lines may naver meet, thus we have to find a new range and calculate again. Using of graphical method is very time consuming and if one step goes wrong, everything will be wrong and you will have to spend much time redo-ing it again. It is not accurate as the answer we get is not inb decimal points and we have to find the answer based on what we see. Parallax error may occur, therefore, this method is not really useful.
SUBSTITUTION METHOD:
Algebra is used in this method. First substitute one variable into another variable before solving the equations.
Substitution may be confusing as the start but once you master the skills of this method, you will find it easier the graphical method as it is not so tedious and time-consuming. Using substitution method, we can get the exact answer more easily. Substitution method is more accurate then graphical method as you are able to work out decimals using this method. It is more convenient then graphical method as we don't have to use a curve ruler to draw the lines and find out the answer. The errors in using this method is definitely not parallex error. The errors that will probably happen is calculation error and misinterpretation of questions.
e.g USE THE SUBSTITUITON METHOD TO SOLVE THE SIMULTANEOUS EQUATIONS:
x+3y=8----------[1]
2x+5y=13----------[2]
From [1], x=8-3y----------[3]
Substituting [3] into [2],
2(8=3y)+5y=13
16-6y+5y=13
16-y=13
y=3
Substituting y=3 into [3],
x=8-3(3)
=-1
therefore the required solution is x=-1, y=3.
ELIMINATION METHOD:
This is a method involving algebra equations too. Elimination is a method whereby you eliminate a particular unknown by adding or subtracting the equations. I think that this method is the easiest to use as you won't go wrong in creatinga new equation like in substitutiong method. This method is very accurate as you can get decimals in this case. In my opinion, this method is more convenient and efficient. I thought that you can get the answer more easily and with much confidence this way.
e.g SOLVE THE SIMULTANEOUS EQUATIONS 5x+2y=18 AND 6x-2y=26 BY THE ELIMINATION METHOD.
5x+2y=18---------[1]
6x-2y=26---------[2]
since the coefficients of y in these equations are +2 and -2 respectively, we add equations [1] and [2] to eliminate the terms of y.
Adding [1] and [2]:
(5x+2y)+(6x-2y)=18+26
11x=44
x=4
Substituting x=4 into [1],
5(4)+2y=18
2y=-2
y=-1
therefore the required solution is x=4 and y=-1.
Lim Qi Min [14]
2A
Lye Jie Yi (16) :)
-Elimination Method
The elimination method is basically using equations to eliminate each other and finding or eliminating an unknown equation. You can then substitute the new known x or y into any of the equations to find the other unknown in the equations and then you get your solution.
Eg.
3x+5y=1---(1)
4x+7y=2---(2)
(1)x4: 12x+20y=4---(3)
(2)x3: 12x+21y=6---(4)
(4)-(3):
y=6-4
=2
Sub y=2 into (1):
3x+5(2)=1
3X+10=1
3x=-9
x=-3
-Substitution Method
The substitution method is also very accurate and convenient, but I think that in most cases, you will have to do more steps in order to solve the equations. You will need to find one of the unknowns first before substituting it in one of the equations to find the other unknown, so it is not as efficient.
Eg.
x+2y=12---(1)
3x-y=1---(2)
From (1), x=12-2y---(3)
Sub(3) into (2):
3(12-2y)-y=1
36-6y-y=1
36-7y=1
-7y=-35
7y=35
y=5
Sub y=5 into (3):
x=12-2(5)
x=2
-Graphical Method
This method of solving simultaneous equations is the most time consuming and not very accurate. The results are obtained when you plot the graph on the graph paper and the point of intersection of the lines for both of the equations will be the solution.
It is time consuming as you will need to plot the lines and etc. which I usually take about 15-20 min to do so. :( It is not very accurate also as, it is not able to give definite and accurate answers like the elimination as substitution method if its in a decimal point or fraction. That’s why I don’t like this method the most! >.<
I generally like the elimination method more as I think that using it will be easier for me to solve an equation. It is more time saving , which makes it more efficient and I think that using it to solve the equations, I would not make as much mistakes. It is also more convenient as you only need to use some of the equations you get when you do the sum along the way to subtract or add each other to get rid of the redundant coefficients or etc. and you can get the answer very quickly. :)
Why learn Simultaneous Equations?
Simultaneous Equations can be used to solve problems like the one mentioned above. It is a better method to use than trial and error, as it is more reliable, simpler and faster
In the ‘elimination’ method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.
Solving simultaneous linear equations in two unknowns by elimination
How to solve
Step 1 } ax+by=c
dx+ey=f
Step 2 Solve the equation for y.
Step 3 Substitute the value of y found into either of the equations.
Step 4 Solve the equation for x.
Solving simultaneous linear equations in two unknowns by substitution
How to solve {ax+by=c (1)
dx+ey=f (2)
Step 1 use one of the equations to express x in terms of y.
Step 2 substitute the new equation into the other equation.
Step 3 solve the equation for y.
Step 4 substitute the value of y into either of the equations.
Step 5 solve the equation for x.
In terms of efficiently and convenience, I think that the elimination method is the best way. There are fewer steps, thus not time-consuming and is less confusing. You only had to make one coefficient in each of the equation the same. Instead of the substitution method which you would find difficult if the question is as follows
Substitution Method
3x-y=7-------- (1)
6x+4y=17-------- (2)
From (2), 4y=17-6x
y=17-6x
4-------- (3)
Sub. (3) into (1),
3x-17-6x =7
4
12x-(17-6x)=28
12x-17+6x=28
18x=45
x=2.5
Sub. x=2.5 into (3),
y=17-6(2.5)
4
=0.5
Elimination Method
3x-y=7-------- (1)
6x+4y=17-------- (2)
(1)*2:
6x-2y=14-------- (3)
(2)-(3):
6y=3
y=0.5
Sub. y=0.5 into (1)
3x-0.5=7
3x=7.5
x=2.5
A man buys 3 fish and 2 chips for £2.80
A woman buys 1 fish and 4 chips for £2.60
How much are the fish and how much are the chips?
First we form the equations. Let fish be f and chips be c.
We know that:
3f + 2c = 280 (1)
f + 4c = 260 (2)
There are two methods of solving simultaneous equations. Use the method which you prefer:
Elimination
This involves changing the two equations so that one can be added/ subtracted from the other to leave us with an equation with only one unknown (which we can solve). We can 'change' the equations by multiplying them through by a constant- as long as we multiply both sides of the equation by the same number it will remain true.
In our above example:
Doubling (1) gives:
6f + 4c = 560 (3)
Since equation (2) has a 4c in it, we can subtract this from the new equation (3) and the c's will all have disappeared:
(3)-(2) gives 5f = 300
∴ f = 60
Therefore the price of fish is 60p
So we can put f=60 in either of our original equations. Substitute this value into (1):
3(60) + 2c = 280
∴ 2c = 100
c = 50
Therefore the price of chips is 50p
Substitution
The method of substitution involves transforming one equation into x = (something) or y = (something) and then substituting this something into the other equation.
So,
Rearrange one of the original equations to isolate a variable.
Rearranging (2): f = 260 - 4c
Substitute this into the other equation:
3(260 - 4c) + 2c = 280
∴ 780 - 12c + 2c = 280
∴ 10c = 500
∴ c = 50
Therefore the price of chips is 50p
Sub. c=50 into (2)
f+4(50)=260
f+200=260
f=260-200
f=60
Therefore the price of fish is 60p
From the above two examples, we can deduce that the elimination method is a shorter method, requiring less steps, compare to the substitution method , it is thus more efficient and less time- consuming.
Now the graphical Method,difficult and troublesome.
Step1:Plot the equations on the graph
Step2:Find the interception point of the graph(the point where the two lines meet/form a cross)
It waste paper(this means that even more trees are dying),time (it is time-consuming ) and needs alot of work(it is tedious).
Generally ,i feel that elimination method is better than substitution method and substitution method is better than graphical method.Only after reading the question would i decide which method i would use to do the sums.
Sorry Ms Chum for only posting now as I didn't knew the date line was on 14 June.
a)The Geographical Method is not very accurate as we may have to estimate most of the possible solution of the equation as it would be very time consuming to search for every possible solution. And there might be problem like human error
It is not efficient at all as it is very time consuming to plot a graph. It is also not efficient at a graphpad or a grid is required to ensure accuracy. The Substitution method is accurate as calcalating is less prone to mistake compared to drawing graph which might be wrong due to parallex error and other human error. The substitution method is efficient as u are only required to look for how much 1 of the unknown is equivalent to the other unknown to solve the equation. However, the elimination method would be more straight foward is certain situation. It is convenient as u would only need a paper and a pen to solve it or if it is simple enough, mental calculation might be enough.to solve it.The elimination method is accurate as it only requires calculation and there is no need to plot any graph. It is efficient as it is straightfoward as all it requires was to make the same unknown to have the same coefficient in both equation and it can be solve easily. As for convenient, it is the most convenient if the 2 equation provides the same coefficient for both the equation as all u have to do is to add or subtract the 2 equation together and solve the linear equation to get the solution for 1 of the unknown.
b)I would usually not recommend using the Graphical Method as it is easy to make mistake if not careful and time-consuming.
The Substitution Method would be recomennded if an unknown is the subject of the equation.
e.g) X=2y+3
This is because all you would need is to substitute the equation into the other equation and solve the linear wquation to solve the simultaneous equation.
The elimination method would be recommended if the 2 equation has the same coefficient
e.g) 2x+3y=8------(1)
5x+3y=11-----(2)
This is because all you need is to subtract 1 equation with the other and solve the linear equation you get to solve the simultaneous equation instead of having to changing the equation to make 1 of the subject an unknown and solve it the substitution method.
Jiayu(15)
There are three methods to solve simultaneous equations. These three methods are graphical method, substitution method and elimination method.
Graphical Method
I do not really like this method as this is not as accurate and efficient than the other two method.To be able to find the answer of simultaneous linear equation using the graphical method, we first have to plot the equation and then find the point where the two lines meet together. This will lead us to the answer to it. But, this may noy be as accurate and efficient compared to the substitution method and the elimination method as we may not draw the line accurately or we may get the wrong answer on the first part. If the two lines never meet, this will cause us to redo our work one more time. Drawing graph and plotting will waste a lot of time compared to elimination method and substitution method. Graphical method are also not accurate in a way that we are not able to get fraction or decimal answer.
Subsititution method
I personally do not really like this method as this method is quite easy to make careless mistakes. It is not efficent in a way that you have to form another equation using the equation given to you so as to use the new equation to substitute into one of the equation. Sometime, you might be careless and use the wrong equation to substitute, resulting in having a wrong answer. You might also have to keep checking your answer and equation as it is quite confusing to have so many equations and this may lead to wasting of time. It will take a lot of time to change the equation, subsitute it, simplify it and solve it. Therefore, it is not efficient.
Example:
x=y+5-----(1)
2x+5y=31------(2)
Sub (1) into (2)
2(y+5)+5y=31
2y+10+5y=31
7y=21
y=3
Sub y=3 into (1)
x=3+5
=8
Solution: y=3,x=8
Elimination method
I prefer this method more than graphical method and substitution method as i think it is the most efficient and accurate method. Although elimination method is almost the same as substitution method, but it is more efficient than it in a way that it wasn't as confusing as substitution method. It is also more efficient and accurate than graphical method as elimination method do not require drawing which will help to save up a lot of time.
Example:
3x-2y=11-----(1)
5x+3y=50-----(2)
(1)*3:
9x-6y=33-----(3)
(2)*2:
10x+6y=100-----(4)
(3)+(4):
19x=133
x=7
Sub x=7 into (1)
3(7)-2y=11
21-2y=11
-2y=-10
y=5
Solution: x=7, y=5
I personally like the elimination method more than the graphical method and substitution method as i think it suit me for i am a careless person as there will be many careless mistakes when you will using substitution method as it have many equations which will make you more confusing, resulting in having many careless mistakes. Elimination method also take up less time than graphical method. Thus, i would use elimination method more when solving simultaneous equation.
Hiies!!! Liyun (02) here!!! Came here to do my homework ):
Everyone whom have posted like write a lot lehs, in fact a lot!!! Hahahaas (:
There are three methods taught. I will talk about the different methods accordingly.
Firstly, I will talk about- (a) GRAPHICAL METHOD
In the aspect of accuracy, I think that graphical method is the most inaccurate way of solving the equations. People tend to draw their graph according to their own scales; causing different people have different scales when drawing the graph. Thus answers may vary according to the different scales used. When drawing graphs, we might not be so lucky to get a perfect number (e.g. 2, 4). At times we do get some weird fractions and decimals and causing us to have a hard time dividing the scales accordingly. Of course, while dividing, we do make some human errors and sometimes, the graphs cannot be divided equally to accommodate the fractions or decimals. Thus we have to estimate and of course, the results produced are not very accurate.
In the aspect of efficiency, I feel that graphical method is not as efficient as the other two methods. When drawing graphs we have to be very careful not to make any mistakes. Once we make a slight careless mistake, it might cause us to get the wrong answer. With the wrong answer, we will often have to redo the graph. But in some other ways, if we do get a perfect number, we can actually get the answer of x and y easily, rather than spending time to solve the equations using the other two methods.
In the aspect of convenience, I personally think that it is the most inconvenience method. Doing the graph, we need the help of graph paper to plot an accurate graph whereas the other two methods can be easily done on any paper. And when doing graph, we need to draw the tables of values, plot the graph and lastly, draw the graph. All of the things that have to be done are time-consuming, it is not easy too. During exam, we might not have the time to lot a graph properly if we do not practice enough. So, graphical method requires a lot of practice.
*No examples to be provided (:*
Secondly, I will talk about SUBSTITUTION METHOD
In the aspect of accuracy, Substitution method is as accurate as Elimination method and thus it is more accurate than Graphical method. Solving simultaneous equation using substitution method allows us to get an exact value of answer rather than estimating the answer. But at times we do make some calculation errors or careless mistakes causing us not to get the correct answer.
In the aspect of efficiency, I personally feel that Substitution method is a rather tedious method to use, there is some specific ways to use this method and one must decide on which x or y to be the subject. Using a method that is long and tedious, thus it allows us to make careless mistakes very easily. But one must be able to handle their algebra well as at times the equation is not a perfect number, and especially fractions, it is not very easy to substitute as it can be very confusing to calculate the actual amount. It can be efficient at times when cannot find the LCM to eliminate. We can easily use this method to substitute, and find the answer.
In the aspect of convenience, it is actually quite convenient as we just need the help of paper (recycled ones also can!!!) and calculator. In some ways, this method is not as convenient as we need to do more practices to get hang of this method. There are some methods to follow, which are simple to understand:
1) Make one unknown the SUBJECT of the equation
2) SUBSTITUDE obtained equation into the other equation.
3) SOLVE for unknowns.
But in some circumstances, if the equation given is in fraction, it is not very easy and convenient for us to solve as we might get confused or make careless mistake easily, which can cause us not to get the correct answer and we can result into losing a lot of marks.
*E.G.*
x+ 2y = 12 ---------- (1)
3x- y = 1 ---------- (2)
From (1), x = 12- 2y ---------- (3)
Sub. (3) into (2), 3(12- 2y)- y = 1
36- 6y- y = 1
36- 7y = 1
35 = 7y
y = 5
When y = 5,
3x- 5 = 1
3x = 6
x = 2
Lastly, I will talk about ELIMINATION METHOD
In the aspect of accuracy, this method has the equal level of accuracy of the substitution method as it can help us get the correct and accurate answer compared to graphical method. We need not estimate the answers like the graphical method. The same thing, we do make careless mistakes if we are not careful enough.
In the aspect of efficiency, I feel that this is the most efficient method. This method is short and simple to understand. For slow learners like me, this method is the best for me to solve simultaneous equations. This method is short and it is not as confusing as substitution method. Thus it is easier to master the techniques to do use this method to solve simultaneous equations. With this method, we need not have to substitute algebra here and there to confuse ourselves. But at times when we get fractions (improper fractions), it is not easy to find the LCM of the two equations, so it give room to careless mistakes.
In the aspect of convenience, this method is convenient as we only need a pen (or pencil), a piece of paper to do and a calculator [Like substitution method]. Because this method is easy to understand, it actually requires lesser practices to be able to understand how to do to with this method. Here are some simple steps to follow:
1) Identify COEFFICIENTS with the SAME NUMERICAL VALUE in the two equations.
2) ADD/ SUBTRACT the equations to ELIMINATE 1st unknown.
3) SOLVE for the 2nd unknown.
4) SUBSTITUTE value of 2nd unknown to SOLVE for 1st unknown.
*E.G.*
5x- 9y = 32 ---------- (1)
3x- 8y = 27 ---------- (2)
(1)x 3: 15x- 27y = 96 ---------- (3)
(2)x 5: 15x- 40y = 135 ---------- (4)
(3)- (4): -27y- (-40y) = 96- 135
-27y+ 40y = -39
13y = -39
y = -3
Sub. y = -3 into (2),
3x- 8(-3) = 27
3x = 27- 24
x = 1
(b) I would depend on the circumstances to decide on which method to use. But of course I would narrow my choices from 3 to 2- Substitution method and Elimination method. If the questions consists of fractions with I will choose to work with the elimination method. It is easier to do as we need not do so much of substituting and confuse ourselves.
EXAMPLE (fraction):
x/6 + y/3 = 4 ---------- (1)
x/12 + 2y/3 = 4 ---------- (2)
(1)x 6: x+ 2 y = 24 ---------- (3)
(2)x 12: x+ 8y = 48 ---------- (4)
(3)-(4): -6y = -24
-y = -4
y = 4
Sub. y = 4 into (3),
x+ 2(4) = 24
x+ 8 = 24
x = 16
But in some circumstances when we are dealing with perfect number, I would choose substitution method.
EXAMPLE:
x+ 3y = 8 ---------- (1)
2x+ 5y = 13 ---------- (2)
From (1), x= 8- 3y ---------- (3)
Sub. (3) into (2),
2(8- 3y) + 5y = 13
16- 6y+ 5y = 13
16- y = 13
y = 3
Sub y = 3 into (3),
x = 8- 3(3)
= -1
If the method is not specified, I will choose in between these two choices. (:
Done by: Chen Liyun (02)
There are three methods in solving simultaneous equations. They are graphical, elimination and substitution method.
GRAPHICAL METHOD:
In graphical method, we have to spend a lot of effort to complete the whole graph before we are able to solve the whole equation. So I do not prefer to use this method as I think that this method is time-consuming. This method may not be very accurate as our lines may not be drawn properly resulting in a wrong answer and in loss of marks. Many times, because of coming to an answer that is not very realistic, we have to spend time and effort to find out what went wrong in our calculations and most importantly our drawing of the graph. This may sometimes cause us to spend too much time resulting in not being able to complete the paper due to lack of time.
SUBSTITUTION METHOD:
For substitution method, we have to first make an unknown of one equation the subject and solve the second equation with that. I think that this method not really very good as sometimes we get fractions and decimals. This result in being unable to be sure whether it is really accurate and we have to be really careful with the numbers. If we are not careful, we may just make a mistake if we are careless. Once we make a mistake we will not be able to solve the second equation and have to look back all the solutions we did and try to figure out the mistake.
ELIMINATION METHOD:
For elimination method, we will first have to make an unknown, AB in the first equation and AC in the second equation the same. Then we will eliminate out the same unknown of the two equations. Thus being able to solve the only unknown left. I think that this method is the most efficient method. As there isn’t so much of the possibility that we will get a fraction or a decimal, since we will most probably multiply the unknowns, we will be able to solve the equations the fastest.
Thus out of these three methods, I would prefer the elimination method most. I feel that using this method has allow me to solve equations the fastest.
Wong Jing Ru
(31)
a)Graphical method, Substitution method, and elimination method are what we have learned. Both three methods could be used to solve simultaneous linear equations. But if want to compare, I would choose the elimination method. Below explains why.
Graphical method
GOOD points about the method..
- (Sadly to say, i really cannot find any reasons of it to be a good method.)
BAD points about the method..
- Requries a lot of time (10mins for me)
- Requries a lot of effort(if you draw the graph wrongly, you need to ERASE and RE-DRAW again.)
- The results are not that accurate.
Firstly, the 'machines' used are a lot. You need pencils, erasers, curve ruler, and a patient mind.
This is because when you draw he graph wrongly, you need to re draw again. Which is a tough work.
Next, even you found the intersection point on the graph, if they lie on the small little boxes, you would need to estimate the answer, resulting the answer to be inaccurate.
----------------------------------
Substitution
GOOD points about the method..
- It gives us ACCURATE answers
BAD points about the method..
- Requries a lot of steps and careful calculations.
- Fractions would appear.
1. label the equations (1) and (2) respectively. followed by (3)and (4).. so on ..
3b - a = 24 ------ (1)
12a + 8b + 4 = 3b - a ------ (2)
2. make one unknown the subject of the equation.
From (2) : 13a + 4 = -5b ----(3)
From (1) : 3b - a = 24
24 = 3b - a
3b = 24 + a
b = (24 + a)/3 ----(4)
3. Substitute obtained equation into the other equation. Then work out normally to solve for unknowns.
Sub (4) into (2),
13a + 5[(24 + a)/3] = 4
(39a/3) + [(120 + 5a)/3] = 4
44a + 120 = -12
44a = -12 - 120
44a = -132
a = -3
Sub a = -3 into (4)
b = [(24 -3)/3]
b = 21/3
b = 7
----------------------------------
At first, i preferred this method compared to the elimination method, as i was too lazy to find out the LCM of the unknowns (method used by elimination). But as i do more of substitution questions, i felt that there are more and more fractions in an equation.
--------------------------------
Elimination method
GOOD points about the method..
- fast and easy way
- you wont get as much careless mistakes as compared using the substitution method
- also very accurate!
BAD points about the method..
(i dont think there is..)
Identify the coefficients with the same numerical value in the two equation.
Then add or subtract the equations to eliminate the 1st unknown.
Next, solve for the 2nd unknown
Then using substitution method, you can find the 1st unknown!
Isn't it simple?
----------------------------------
b)I would choose elimination out of the three..
At first, i preferred substitution method compared to the elmination method, as i was too lazy to find out the LCM of coefficients pf the unknowns (method used by elimination). But as i do more of substitution questions, i felt that there are a lot of fractions in an equation. Hence i was messed up by them. Thus i decided to use the elimination method to solve a system of simultaneous linear equations.
------------END------------------
Ong Yet Lin (18)
Leona here
*Sorry Ms Chum, I didn’t know that the deadline was over…
Luckily you postponed the deadline…*
Well, let me start here now…
The 3 methods we have learnt for how to solve simultaneous linear equations are the graphical, substitution and elimination method.
Let me start with the graphical method.
Basically, every point that is plotted on the graph of each equation is a solution of the equation; therefore each equation has an infinite number of solutions using the graphical method. But a solution for a simultaneous linear equation in two unknowns, for example x and y, needs to be a pair of values that x and y can satisfy both equations. If there is a solution, it will be the coordinates of the point of intersection of the graphs of the simultaneous equations. When two simultaneous equations are solved graphically, the solution obtained may be an approximation only. For example, it is not possible to read the exact solution if the answer is a fraction, thus the graphical method is not very accurate. It is also very time-consuming and we will prefer to avoid using this method during exams if possible.
(sorry, no examples are available, as I could not load graphs into the computer and post.)
Moving on to the next method we learnt, the substitution method.
The substitution method involves algebraic expressions as it express its terms by transforming one equation into x = (something) or y = (something) and then substituting this new answer into the other equation, to remove one unknown, then solving for another.
An example will be:
3x + 2y = 280----------- (1)
x + 4y = 260------------ (2)
Rearrange one of the original equations to isolate a variable.
Rearranging (2): x = 260 – 4y
Sub. (2) into (1),
3(260 – 4y) + 2y = 280
780 – 12y + 2y = 280
780 – 10y = 280
10y = 780 – 280
= 500
y = 500/10
∴ y = 50
Then, Substitute y=50 into one of the original equations to get
x + 4(50) = 260
x + 200= 260
∴ x = 260 – 200
= 60
∴ x = 60 and y = 50
When two simultaneous equations are solved using the substitution method, the solution obtained will be exact, hence I could say that it is accurate. It is not as time-consuming as the graphical method therefore it is efficient and convenient to use, although it is a little long, but it gives accurate answers.
Lastly, I will write about the elimination method.
The elimination method involves the changing of two equations so that one can be added/ subtracted from the other to leave us with an equation with only one unknown (which we can solve). We can 'change' the equations by multiplying them through by a constant as long as we multiply both sides of the equation by the same number.
Using the same example I have written above,
3x + 2y = 280----------- (1)
x + 4y = 260------------ (2)
(1) x 2: 6x + 4y = 560----------- (3)
Since in both equations (2) and (3) have 4y, we can subtract this from the new equation (3) and the y will all have disappeared, leaving us with the x to solve first.
(3) – (2): 5x = 300
x = 300/5
∴ x = 60
Now, put x = 60 in either of the original equations.
Sub. x = 60 into (1),
3(60) + 2y = 280
180 + 2y = 280
2y = 280 – 180
= 100
y = 100/2
∴y = 50
∴ x = 60 and y = 50
When two simultaneous equations are solved using the elimination method, the solution obtained will be exact, thus I could say that this method is accurate. It is not as time-consuming as the graphical nor the substitution method therefore I find it the most efficient and convenient method to use.
I will choose to use the elimination method to solve a system of simultaneous equations if the method is not specified in the question, is because, when comparing the accuracy, it is better than the graphical method, and I find that the elimination method would take lesser steps then in the substitution therefore it would help me save time. Using the above example, the elimination method uses 8 steps, when the substitution method uses 10 steps, that shows that the working needed to do the elimination method is shorter than the substitution method, even if it is shorter by a few steps, that also consumes time and in exams, the elimination method would get us to the answer faster than the substitution method, saving us a few more minutes for the rest of the paper.
LaukokTing (36)2A
I think i would prefer Elimination method. I would explain why i dont like the other 2.
First the graphical method, i think this is very time consuming and not accurate. In test when time is tight, using graphical method is not wise as we need alot of time to draw the graph and also to calculate the points x and y. If we make a mistake in the calculations, the whole graph will be wrong and therefore we will be wasting our precious time correcting our mistake. Therefore in conclusion, i think graphical method is a no-no as it is tedious, time consuming and inaccurate.
I think substitution method is also not a wise choice. Substitution method is sometimes very complicated when the fraction and decimals come in. We get weird numbers and equations when we can then solve with the help of calculators. Thus if there are no calculators, we will be also spending alot of time solving. In conclusion, i think substitution method is also time consuming, complicated and tricky.
However i think the elimination method is easier because we only need to eliminate the same coordinates. It is easy, time saving and not so tedious. We only need to mutiply and make both coordinates equal. Then we can eliminate and thussolve a question easily. In conclusion i think elimination method is the best way to use because it is time saving, non-tedious and easy to use.
There are three ways to solve simultaneous linear equations in two unknowns. They are the graphical method, substitution method and the elimination method. These are three very different ways to solve equations. I would personally prefer to use the elimination method to solve simultaneous linear equations as it is more accurate than the other two methods and it is also easier for me to solve the equations given. Allow me to explain why is this so (:
GRAPHICAL METHOD (:
A system of two linear equations is called simultaneous linear equations in two unknowns x and y. A solution of this system is a pair of values of x and y that satisfy both equations. Hence if a solution exists, it is the coordinates of the point of intersection of the graphs of the simultaneous equations. When using the graphical method, one drawback is that we may not get the exact solution of the equation sometimes. We humans do make mistakes sometimes; we may calculate the answer wrongly, resulting in the wrong answer. Graphs are a lot harder to calculate than the other two methods which are much easier, efficient and accurate.
SUBSTITUTION METHOD (:
The substitution method requires algebra to obtain the exact solution. To solve simultaneous equation using the substitution method, you can begin by expressing y in terms of x or vice versa.
For example,
x + 3y = 8
2x + 5y = 13
Solution:
x + 3y = 8 ---------- (1)
2x + 5y = 13 ---------- (2)
From (1): x = 8 – 3y ---------- (3)
Substituting (3) into (2):
2(8 – 3y) + 5y = 13
16 – 6y + 5y = 13
16 – y = 13
y = 3
Substituting y = 3 into (3):
x = 8 – 3(3)
x = -1
Thus, the required solution is x = -1 and y = 3.
ELIMINATION METHOD (:
In certain cases, it is more convenient to solve a system of simultaneous linear equations by eliminating a particular unknown by adding or subtracting the equations. This technique is called the elimination method.
For example,
5x + 2 y = 18 ---------- (1)
6x – 2y = 26 ---------- (2)
Adding (1) and (2):
(5x + 2y) + (6x – 2y) = 18 + 26
11x = 44
x = 4
Substituting x = 4 into (1):
5(4) + 2y = 18
2y = -2
y = -1
Thus, the required solution is x = 4 and y = -1.
In some cases, we may need to eliminate one unknown, say, x, by making the coefficients in both equations to be the LCM of the given coefficients. The advantage of using LCM instead of just the product of the coefficients is that the values of the coefficients in the resulting equations are smaller.
For example,
3x + 5y = 1
4x + 7y = 2
Solution:
3x + 5y = 1 ---------- (1)
4x + 7y = 2 ---------- (2)
(1) x 4: 12x + 20y = 4 ---------- (3)
(2) x 3: 12x + 21y = 6 ---------- (4)
(4) – (3):
(12x + 21y) – (12x + 20y) = 6 – 4
y = 2
Substituting y = 2 into (1):
3x + 5(2) = 1
3x = -9
3 = -3
Thus, the required solution is x = -3 and y = 2.
As the elimination method is much easier, convenient, and efficient and last but not least, more accurate to use, I would personally prefer to use the elimination method to solve simultaneous linear equations. Thank you!!! (:
Eugenia Teh Shu Wen (4)
2A
Of all 3 of the simultaneous equations which we can use, i prefer to use the Elimination Method as compared to the Subtitution Method and the Grahpical Method.
1st of all, i dislike using the Graphical Method as i feel that it is very time consuming and isnt very accurate when it comes to question which have decimals or fractions.
i prefer to use the elimination method as it is the simplest method which involves removing or eliminating one of the unknowns to leave a single equation which involves the other unknown.
Example:
Solve the simultaneous equations
3x + 2y = 36 (1)
5x + 4y = 64 (2)
Solution:
Notice that if we multiply both sides of the first equation by 2 we obtain an equivalent equation
6x + 4y = 72 (3)
Now, if equation (2) is subtracted from equation (3) the terms involving y will be eliminated:
6x + 4y = 72 − (3)
5x + 4y = 64 (2)
x + 0y = 8
So, x = 8 is part of the solution. Taking equation (1) (or if you wish, equation (2)) we substitute
this value for x, which will enable us to find y:
3(8) +2 y = 36
24 +2 y = 36
2y = 36− 24
2y = 12
y = 6
And then there is the method of substitution:
Solve one of the equations for one unknown in terms of the other.
Then, substitute that in the other equation.
That will yield one equation in one unknown, which we can solve.
Let us solve equation 1) for y:
1) y = 4 − 2x
And now, substitute this for y in equation 2):
2) x − (4 − 2x) = −1
This equation has only the unknown x:
x − 4 + 2x = −1
3x = −1 + 4
3x = 3
x = 1
To find y, substitute x = 1 in line 1):
y = 4 − 2· 1
y = 2
We have learnt the three simultaneous linear equations, which is the graphical, substitution and elimination methods. However I would prefer the elimination method as less time would be wasted. Let us compare!
Graphical method:
Many people thinks that this is a very time consuming method, so do I.To get the correct answer, you have to be sure that you plot the graph correctly. First, you will need to create a table of x from 1 - 6. After that you can find the value of y. Next, plot the graph accordingly. Once you see the solution that satisfy both equation, that will be your answer. The efficiency and is there as we can straight away look at the answer. It is not very convenient to use this method as it takes time to draw the graph and more time have to be spent to check for mistakes. If the answer is a fraction, we may not necessarily get the accurate answer.
Substitution method:
This method is a more reliable one, accuracy-wise. First we must use an equation and find the unknown(x). After finding the unknown, then we can substitute the second equation into the third. Continue doing and we will get the required solution. It is effective as the answer turns out to be accurate. Although it is fast, the final and third method, turns out to be even more convenient.
Elimination method:
This method involves the need to times one of the equation, so that the similar coordinates can be added or substracted, in this case "Eliminated".
Therefore in my opinion i think the elimination method is the best way in the 3 aspects. So when solving a system of simultaneous linear equations, if the method is not specified, i'll use the elimination method without any doubt. This not only saves time, and will also ensure that the answer is accurate. However we mustn's make any careless mistakes, or we will miss out on getting some points in examination.
Like most others, i prefer elimination method than the rest. This is because elimination takes a faster time. Also the result it produce is more accurate. Let us look at one example of elimination method.
3x - 5y = 10 --- (1)
4x + 7y = 150 --- (2)
(1) x 4: 12x - 20y = 40 --- (3)
(2) x 3: 12x + 21y = 450 --- (4)
(4) - (3): 41y = 410
y = 10
Sub y = 10 into (1),
3x - 5(10) = 10
3x - 50 = 10
3x = 60
x = 20
As we could see, we can just times the equations to get 2 new equations, then subtract the coefficient to get the solution to y. This is very effective and takes less time than the substitution method.
The other two method are graphical and substitution method.
For graphical method, although it is easy to spot the answer, we must be extra careful when plotting the graph. Any careless mistakes and points will be wasted. Also, plotting the graph is time-wasting. So it isnt a very good method to use during exams, as we tend to be more nervous and mistakes can be done. The answer is also not accurate if its a fraction.
For substitution method, it is very much different from the graphical method, and much more like the elimination method. In this method, you have to find the unknown x. Then after this you will need to substitute the second equation into the first one. It is much more accurate than the graphical method. However i think it loses elimination method in the case of time. Like the elimination method, it nonetheless is also effective.
I'll show an example of the substitution method:
x + 5y = 500 --- (1)
3x + 2y= 460 --- (2)
x = 500 - 5y --- (3)
3(500-5y) + 2y = 460
1500 - 15y + 2y = 460
1040 = 13y
y = 80
sub y = 80 into (1),
x + 5(80) = 500
x + 400 = 500
x = 100
With this, i shall end by saying elimination is the easier one as it is easier to times than to substitute.
There are three common methods of solving simultaneous equations. We shall introduce two methods here: the elimination method and substitution method. The third method, the graphical solution method, will be dealt with in curve sketching.
Substitution method:
Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Find an equation that can be rearranged for one variable, that is, it can be rewritten in the form VARIABLE = EXPRESSION, in which the left-hand side variable does not occur in the right-hand side expression. Next, substitute that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved, substitute the solutions found for the variables in the above right-hand side expression.Steps to follow when solving equations using substition method:
Step 1: Make 1 unknown the subject of the equation
Step 2: substitute obtained equation into the other equation
Step 3: solve for unknowns
Example:
Solve the simultaneous equations:
7x - 2y = 21, 4x + y = 57
7x - 2y = 21 --------(1)
4x + y = 13 --------(2)
From (2): y = 57 - 4x --------(3)
Substitute (3) into (1):
7x - 2(57 - 4x) = 21
7x - 114 + 8x = 21
15x = 114 + 21
15x = 135
x = 9
Substitute x = 9 into (3):
y = 57 - 4(9)
y = 57 - 36
y = 21
Solution: x = 9, y = 21.
Elimination method:
Elimination by judicious multiplication is the other commonly used method to solve simultaneous linear equations. It uses the general principles that each side of an equation still equals the other when both sides are multiplied (or divided) by the same quantity, or when the same quantity is added (or subtracted) from both sides. In multiplication/division, a factor is chosen so that when both sides have equivalent quantities added from another equation in the system (that is, the equations are added), one or more of the variables disappear, the resulting equations are still valid representations in the system, and their smaller number of remaining unknowns thus makes the system of equations easier to solve. As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable. Typically, each "back-substitution" can then allow another variable in the system to be solved.
Example:
Solve the simultaneous equations:
13x - 6y = 20, 7x + 4y = 18
13x - 6y = 20 --------(1)
7x + 4y = 18 --------(2)
The coefficients of y in both equations will be numerically equal if we multiply (1) by 2 and (2) by 3, since the LCM of 6 and 4 is 12.
(1) x 2: 26x - 12y = 40 --------(3)
(2) x 3: 21x + 12y = 54 --------(4)
(3) + (4): 47x = 94
x = 2
Sub. x=2 into (1):
13(2) - 6y = 20
6 = 6y
y = 1
Ans: x = 2, y =1.
Graphical mathod:
The first step we have to do is to plot tables for the 2 equations given, followed by ploting the graph. The coordinates where the graphs intersect will be the solution of the simultaneous equations.
So, i wil decide to use elimination method to solve a system of simultaneous linear equations if the method is not specified.The reasons are stated below.
Substitution method: Sometimes, if the equation is given in fraction, it will not be easy for us to solve as we may make careless mistakes easily, which can lead to wrong answers and a loss of marks.
Elimination method: I think this is the most efficient method among among all the three methods. This method is quite accurate, efficient and is rather convenient compared to the other two methods.
Graphical method: To me, the graphical method is both inconvenient and time-consuming. If there is a point where the results we get from the table is a fraction or recurring decimal, it will be difficult for us to plot the graph on the paper and thus we may get an inaccurate answer. During exam, we might not have enough time to plot a graph properly if we do not practice enough. So, graphical method will not be my choice if the method is not specified.
Done by: Jaslyn Tay(8)
I will start by comparing the three methods (Graphical, substitution and elimination) of solving simultaneous linear equations.
Graphical method
Solving simultaneous linear equations by graph requires a lot of time and effort. Firstly, we need to have a table of values of the coefficients. We have to calculate the values for the equation. As we do not need to write down the steps, we might make a mistake without knowing. When we make a mistake in the table, we are bound to be wrong in the graph that we plotted according to the table. Sometimes when we forget to label the graph or write the x and y, we will result in losing marks. The drawing of the graph also waste a lot of time as there is a lot of things to draw. Last but not least, the graph can only show the approximate answer of the equation due to the limit in the boxes thus we cannot get the exact answer.
Substitution method
Unlike graph, substitution method can allow us to get the exact answer to the simultaneous linear equation. It is faster than graph and we can check our steps for errors. However, we still need a bit of time. This method requires a lot of steps. Firstly, we need to rearrange the equation to make one of the unknowns the subject of the new equation. Then, with the new equation, we have to substitute the new equation into the other equation. Only then have we found one of the unknowns. With the answer, we would still need to substitute it into one of the equations to find the other unknown.
Example :
2x+y=25
3x-4y=-1
Solution :
2x+y=25 --- > (1)
3x-4y=-1 --- > (2)
From (1) : y= 25-2x --- > (3)
Substitute (3) into (2) :
3x-4(25-2x)=-1
3x-100+8x=-1
11x=99
x=9
Substitute x=9 into (3) :
y=25-2(9)
=7
The solution is x=9, y=7.
Elimination method
In certain cases, it is more convenient to solve a system of simultaneous linear equations by eliminating a particular unknown by adding or subtracting the equations. Elimination, too, requires lesser time than graph. It also reduces the possibility of making mistakes as the steps are written down and we could easily check for mistakes. Elimination also allows us to get the exact answer to the equations thus the accuracy is higher than graph. Elimination is more convenient than substitution. We could easily eliminate away a coefficient using elimination. With that, it would make things easily to solve the whole equation.
Example :
2x+3y=3
x-3y=6
Solution :
2x+3y=3 --- > (1)
x-3y=6 --- > (2)
(1)+(2) : 3x=9
x=3
Substitute x=3 into (1):
2(3)+3y=3
6+3y=3
3y=-3
y=-1
The solution is x=3, y=-1.
Elimination is the most efficient method as it requires less time and reduces the possibility of making mistakes due to the changes in the signs when carried over to the other side.
How do you decide which method to use to solve a system of simultaneous linear equations if the method is not specified?
I would start by forming the equations. I would not consider using graph as it requires a lot of time. If one of the equations show one of the coefficient as the subject of the equation, like for example y=50-2x and 3x-4y=-1, I would choose to use the substitution method. I would not need to form another equation thus substitution method is chosen. If the equation does not show one of the coefficient as the subject of the equation, like for example 2x+3y=6 and 5x+9y=12, I would choose to use the elimination method as I could easily eliminate away the y.
Sim Hui Ping (21)
Miss Chum, sorry to hand up 5 days late but I really don't have time to do. :D
In this journal, we are supposed to compare 3 different methods, naming Graphical method, Substitution method and Elimination method.
First, we will start with Graphical method.
In an example:
x + y = 6
x - 2y = 3
A solution of this system is a pair of values of x and y that satisfy both equations. Hence if there is a solution; it is the coordinates of the point of intersection of the graphs of the simultaneous equation.
There may be benefits and disadvantages when using graphical method. For me, I believe it is easier to understand if we draw out the graph because by just looking at it, we could see the solution clearly which is the part where it intercepts.
However, it can be very troublesome. Firstly, we must have a table of values. Next, according to the values, we must plot two lines and examine the part where it intercepts. Lastly, we must record down the coordinates of the point of intersection. By doing so, it takes a very long time. If we make a small mistake when plotting, there is a high chance that we need to redo it over again. Furthermore, the solution obtained may be an approximation only as it is impossible to read the exact solution, thus making it the last choice to use during examination.
Now, to Substitution Method. (:
Substitution Method, together with elimination method is example of algebraic method. Both of them can enable us to obtain exact solution. Definitely, Substitution method is a better choice than Graphical method but it, also, take quite a long time as it has many steps thus making it challenging and confusing.
Firstly, we must rearrange the equation such that either an unknown is the subject of an equation. Secondly, we must substitute this equation into the second equation and by doing so, we had got the solution for one unknown. Lastly, we can substitute the solution into the equation, getting the solution for the other unknown. Finally, we got the answers for both unknown.
Example 1:
3x - y = 7
6x + 4y = 17
3x - y = 7 ……………… ( 1 )
6x + 4y = 17 …………. ( 2 )
From ( 2 ), 4y = 17 – 6x
y = ( 17 – 6x ) ÷ 4…………..( 3 )
Substituting ( 3 ) into ( 1 ),
3x - (( 17 – 6x ) ÷ 4 ) = 7
12x – ( 17 – 6x ) = 28
12x – 17 + 6x = 28
18x = 45
x = 2.5
Substituting y = 2.5 into ( 3 ),
y = (17 – 6 ( 2.5)) ÷ 4
= 0.5
Thus, the required solution is x = 2.5 and y = 0.5.
By looking at the above example, we can see that it took many steps and rather confusing at the same time. Thus, for this type of question, elimination is always preferred to use. However, there is one type of simultaneous equation which substitution will handle faster than elimination.
This type of simultaneous equation has one unknown that has one coefficient. For example, x has one coefficient while 2x don’t.
Example 2:
X + 3y = 8 ( Notice that the x has only 1 coefficient )
2x + 5y = 13
X + 3y = 8 …………….. ( 1 )
2x + 5y = 13 …………. ( 2 )
From ( 1 ), x = 8 – 3y …………..( 3 ) ( the x has only 1 coefficient so we do not need to divide it like example 1 )
Substituting ( 3 ) into ( 2 ),
2 ( 8 – 3y ) + 5y = 13
16 – 6y + 5y = 13
16 – y =13
y= 3
Substituting y = 3 into ( 3 ),
x = 8 – 3 ( 3 )
= -1
Thus, the required solution is x = -1 and y = 3.
Therefore, if it has one coefficient, substitution method is faster.
Now, on to elimination method, the ideal method that everyone like to use.
In certain cases, it is more convenient to solve a system of simultaneous linear equations by eliminating a particular unknown by adding or subtracting the equations.
Example 3:
5x + 2y = 18 ………….. ( 1 )
6x – 2y = 26…………… ( 2 )
Since the coefficients of y in these equations are +2 and -2, we can add both equations together and eliminate the terms.
Adding ( 1 ) and ( 2 ):
( 5x + 2y ) + ( 6x – 2y ) = 18 + 26
11x = 44
x = 4
Sub x = 4 into ( 1 ),
5 ( 4 ) + 2y = 18
2y = -2
y = -1
Thus the required solution is x = 4 and y -1.
By comparing these with substitution method, we can see it take a lesser number of steps. Therefore, it would not be too challenging and confusing. With these advantages, we can complete simultaneous equation in examination more quickly and easily.
On the other hand. in some cases, we may need to eliminate an unknown, say, x, by making the coefficients as both equation s to be the LCM of the given coefficients. The advantage of using LCM instead of just the product of the coefficients is that the values of the coefficients in the resulting equations are smaller.
Example 4 :
3x + 5y = 1 ……….. ( 1 )
4x + 7y = 2 ……….. ( 2 )
( 1 ) x 4 : 12x + 20y = 4 ………..( 3 )
( 2 ) x 3 : 12x + 21y = 6 ………..( 4 )
Since the coefficients of x in equation ( 3 ) and ( 4 ) are both 12, we subtract the equation (3 ) from the equation ( 4 ) to eliminate the terms in x.
( 4 ) – ( 3 ) : ( 12x + 21y ) – ( 12x + 20y ) = 6 – 4
y = 2
Substituting y = 2 into ( 1 ), 3x + 5 ( 2 ) = 1
3x = -9
x = -3
Thus, the required solution is x = -3 and y = 2.
After going through all three of the method, I will give a short summarizing on their advantages and disadvantages =)
Graphical Method:
Advantage:
- Easier to understand
Disadvantage:
- Take a long time
- Usually untidy
- solution not exact
- not ideal when examination
Substitution Method:
Advantage:
- lesser time required than graphical
- exact solution
- some equation are better to solve when Substitution Method is used
Disadvantage:
- quite of number of steps
- confusing
- challenging
Elimination Method:
Advantage:
- lesser time required than graphical and substitution
- exact solution
- take quite fast on all type of question
- less number of step than substitution
- ideal for exams
Disadvantage:
- confusing
- challenging
Done by: Yvonne (29)
okay I answer the first question about which method is the best okay:
I think ELIMINATION METHOD is the BEST (tadah!)
graphy method is so CHIM, its very confusing and tedious cos you gotta draw the graphy which is soooo irritating, sometimes you draw wrongly and wah lao, wrong liao, den you sian right? plus its so not accurate, according to the decimal point thingy…..and its inconvenient. I dun like graphy method ha!
substitution method is so FORGETABLE. true marh, there was this time in a test whereby you HAVE to use that damn method and I just didn’t know how to do it… so I totally lost my mark! sobs lurhs! damn sian right, and its very irritating and prone to mistakes, for me larh, I dunno about the others…
elimination is the best, baby! its so easily remembered! hahaa, I personally prefer that one as I find it easy to eliminate than substitute, its psychological marh! also it makes less mistakes as its not so confusing…
second question answer:
I will choose the elimination method, duh! I like it ok ok… its less time consuming cos substitution need to shift the algebra here and there… so irritating, at least elimination can just totally “minus” the whole algebra thingy which is much faster! hahaa! see ah example barh:
substitution method
use the substitution method to solve the simultaneous method
x + 3y =8 ---------------- method 1
2x + 5y =13 -------------- method 2
from (1), x = 8- 3y -------------- method 3
substituting (3) into (2)
2( 8- 3y) + 5y= 13
16- 6y + 5 y= 13
16- y= 13
y= 3
substituting y =3 into (3)
x= 8- 3(3)
= -1
solution is x =-1 and y=3
elimination method
use the elimination method to solve the simultaneous method
x + 3y =8 ---------------- method 1
2x + 5y =13 -------------- method 2
method 1 x 2 :
2x + 6y= 16----------- method 3
(3)-(2)
y= 3
sub y=3 into method 1
x + 3(3)= 8
x=-1
solution is x =-1, y=3
seeee! the elimination method saves time yeah!!! its better^^
seah xin lu ( 20 )
A) Personally , i felt that using the simiultanueous linear equations by the graphical is the most tedious and time consuming compared to the subsitution and elimination methods . When in exam , you definitely cannot apply the simultaneous linear equations by the graphical method when solving a question as you may run out of time and it is certainly not convenient as well . How are you going to find yourself a graph paper in the midst of an exam . However , during examination , you can certainly apply the subsitution and eliminaion method as they are accurate , efficient and convenient in a way that you do not have to trouble yourself to find a piece of graph paper .
Subsitution is just by using the same , example , make the coefficients of y in equations (4) and (5) the same . In Elimination method , Eg : 5x+2y=18....(1)
6x-2y=26......(2)
Since the coefficients of y in these equaions are +2 and -2 respectively , we add equations (1) and (2) to eliminate the terms in y . They both are somewheat similar .
B) I always use the Elimination method as it decreases the number and makes things look easier . As I am those muddle-head type , when there are many numbers and equations , I will be bound to left out some of the equations and thus , resulting in an unaccurate answer .
Elimination method
5x+2y=18.......(1)
6x-2y=26.......(2)
Adding (1) and (2)
(5x+2y)+(6x-2y)=26+18 ( the unknown y is eliminated )
by eliminating y , this made the question seems clearer and easier to solve .
Subsitution method
5x+2y=18.........(1)
6x-2y=26.........(2)
From (1) , 5x=18-2y
x=18-2y/5 (3)
subsitute (3) into (2) ,
108-12y/5-2y=26 (4)
The subsitution method definitely looks much more complicated than the Elimination method and I would be very careless as to left out some numbers like i always did .
Therefore , I always decided to use Elimination Method to solve a system of simultaneuous linear equations if the method is not specified .
Simone(23)
There are 3 methods to solve linear equations, they are
graphical, substitution and elimination methods.
Substitution method:
Find an equation that can be rearranged for one variable, in which the left-hand side variable does not occur in the right-hand side expression. Next, substitute that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved substitute the solutions found for the variables in the above right-hand side expression.
Elimination method:
It uses the general principles that each side of an equation still equals the other when both sides are multiplied by the same quantity, or when the same quantity is added from both sides. In multiplication/division, a factor is chosen so that when both sides have equivalent quantities added from another equation in the system, one or more of the variables disappear, the resulting equations are still valid representations in the system, and their smaller number of remaining unknowns thus makes the system of equations easier to solve. As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable. Typically, each "back-substitution" can then allow another variable in the system to be solved.
Graphical method:
The first step we have to do is to plot tables for the 2 equations given, followed by ploting the graph. The coordinates where the graphs intersect will be the solution of the simultaneous equations.
So, i will choose either substitution or elimination method if the method is not specified as both are easier methods to me. Graphical method waste alot of time when plotting the graph. While the elimination and substitution method do not waste so much time and both are easy and fast methods when solving simultaneous linear questions.
I prefer elimination.=D
When using elimination,it is more accurate and less prone to careless mistakes,unlike substitution.Sometimes the numbers might have decimals,and when you substitute inside,unless you are very accurate and the question is not very difficult,you will most likely to make mistakes.
When using graphical method,it's very easy as u juz need to point out the points on the graph and it will be correct,but when one point goes wrong,everything will go wrong.But when it comes to time,its very time consuming,with all the drawings and accuracy.
Elimination needs the least time and its easy and accurate for me.It is also less mistake prone,and all the students like it~
This is Joel yeo here.
I would prefer to use the substitution and elimination method compared to the graphical one. Depending on the situation, i would choose one out of the two.
Plotting graph may seem easy but it is not. Mistakes are often done during the process. Thus it is not a very good method.
Substitution is an easier method as we can sub x(for example) from method one into method two and is able to find y(for example) immediately.
Eg. x=2y----(1)
3x=5y+2-----(2)
Sub. x=2y into (2),
6y=5y+2
y=2
Sub y=2 into (1),
x=4
Elimination can be useful when the integer (coefficient)cannot be divided(10x=1).
Eg. 3x-2y=1-----(1)
9x-5y=2------(2)
(1)x3:9x-6y=3-----(3)
(3)-(2):-y=1
y=-1
Sub y=-1 into (1),
3x=-1
x=-1/3
As both methods have their own strengths and weaknesses, i would choose each method depending on the situation.
Gerald Tan (38)
As many of us know, there are 3 types of methods to solve simultaneous equations and they are GRAPHICAL METHOD, ELIMINATION METHOD , SUBSTITUTION METHOD. In aspects of accuracy, the graphical method is the most inaccurate one as we as students can only get a rough figure of the solutions of the equations and no exact answer can be concluded as the equation is presented in a graphical way. Hence, it is neither a efficient method to use nor is it convenient for students to use. In my opinion, i feel that both the substitution method and the eliminaton method are about the same . Hwever , the elimination method is still the easier and moer convenient method among students as they can get a better understanding by using the method. for example,
x+2y=12-----(1)
3x-y=1-----(2)
By using one of the methods,i can solve the equation out by finding out the unknown values of x and y. However , before proceeding to attempt the question, i will always decide which method is the best for me to use. I would observe if one of the equations can be incooperated the other. For the above question, Using the elimination method would be faster.
Solving by the elimination method:
x+2y=12------(1)
3x-y=1-------(2)
(1)X(3):
3x+6y=36----(3)
(3)-(2):
7y=35
Hence,y=5
Sub y=5 into (1):
x+2(5)=12
x+10=12
x=12-10=2 so x=2
However, if i use the subsitution method,more time and steps will be taken instead.
THE SUBSITUTION METHOD:
x+2y=12---(1)
3x-y=1---(2)
x=12-2y---(3)
Sub (3) into (2) :
3(12-2y)-y=1
36-6y-y=1
36-7y=1
35=7y
Hence, y=5
Sub y=5 into (3)
x=12-2(5)
x=12-10
=2
I would leave the graphical method last as it is the most unclear way and most inconvenient way to solve an equation as a definite answer is hard to obtain through this method.
Gan She Hwa (6)
I prefer the elimination method to the other two methods, as I am more familiar with it. It is easier to calculate as it only involves simple calculation, which can be done mentally. Also, elimination method are more time-saving compare to the other two since more time will be spent on drawing the graph (graphical method) and there are more steps to work on the variables (substitution method). However, substitution method and graphical method would be of use based on different circumstances.
Substitution Method
1) Definition
With the substitution method, we solve one of the equations for one variable in terms of the other, and then substitute that into the other equation to solve.
2) Steps
Step 1: Label the equations
Step 2: Make one of the unknowns from one equations the subject
Step 3: Substitute the subject into the other equation
Step 4: Solve for the unknown
Step 5: Substitute the found unknown to either one of the equations to find other unknown
3) Example
X + Y = 10 --------- Equation 1
2X - 3Y = 5 --------- Equation 2
From equation 1,
X + Y = 10
X = 10 - Y
Substitute X = 10 – Y into equation 2
2(10 - Y) – 3Y = 5
20 – 2Y - 3Y = 5
-5Y = 5 – 20
-5Y = -15
Y = 3
Substitute Y = 3 into equation 1
X + 3 = 10
X = 7
Thus, the value of X and Y is 7 and 3 respectively.
4) Accuracy, Efficiency and Convenience
Accuracy – it is less accurate than the elimination method to the graphical method. In my opinion, I will rank the substitution method in the 2nd place in terms of accuracy. As the process involves a lot of subtracting and adding, we will have a higher chance of making careless mistakes. Also, we need to form our own equation to solve the values of the variables; the mathematical sign would confuse us and result in a loss of marks. Personally, I find it quite hard to use substitution method when there is fractions or decimals.
Efficiency- it is less efficient than the elimination method to the graphical method. In my opinion, I will rank the substitution method in the 2nd place in terms of efficiency. It is more tedious to do substitution method as there are a lot of steps and we have to form our own equation. Also, we would need to take note the equations and the answer at all times or otherwise, we might make careless mistakes. If we find that our final answer is wrong, we might have a hard time to calculate the sums and equation, thus, wasting time since we have to make one of the unknowns the subject, substitute into another equation, solve for the value and substitute again to find for the other unknown.
Convenience- it is not convenience as the calculations are not easy to do and there might be careless mistakes. Also, it is not useful to apply this method to daily lives since the process is too tedious and the steps needed is too long.
Elimination Method
1) Definition
It requires the elimination of one of the variables in order to get the values and solve for the other variable.
2) Steps
Step 1: Label the equations
Step 2: Make one of the unknowns from either equation to the same coefficient
Step 3: Eliminate the unknown by either adding or subtracting both equations
Step 4: Solve for the remaining unknown
Step 5: Substitute the found unknown to either one of the equations to find the other unknown
3) Example
X + Y = 10 --------- Equation 1
2X – 3Y = 5 --------- Equation 2
Equation 1 x (2)
2X + 2Y = 20 --------- Equation 3
Hence to eliminate away “2X”, we need to subtract equation 3 from equation 2.
(2X – 3Y) – (2X + 2Y) = 5 – 20
2X – 3Y – 2X – 2Y = -15
-5Y = -15
Y = 3
Substitute Y = 3 into equation 1
X + 1(3) = 10
X + 3 = 10
X = 7
Thus, the value of X and Y is 7 and 3 respectively.
4) Accuracy, Efficiency and Convenience
Accuracy- it is the most accurate method of all. In my opinion, I will rank the elimination method in the 1st place in terms of accuracy. The elimination method has lesser steps, thus shortening process and minimise the risk of careless methods. Also, the calculation is very simple as it is just adding or subtracting the coefficients and the sums would be easier to solve since the digits are smaller. We would not need to estimate or round off the answer like substitution method.
Efficiency- it is the most efficiency method of all. In my opinion, I will rank the elimination method in the 1st place in terms of efficiency. The elimination method is more timesaving and thus helps us to complete tests and allowing us to check our answers. Also, when we come across algebra or decimals, it is easier to calculate the values of the variables easily since we eliminate out one of the unknowns, and at the same time, to minimise confusion.
Convenience- it is very convenience since the calculations can be done mentally and we do not need to bring a calculator around at all times. The steps needed are short and thus, we can find the most accurate answer in the shortest time possible.
Graphical Method
1) Definition
It requires the finding of values of the unknowns from both equations through drawing lines and then takes the point that the two lines intersect as the answer.
2) Steps
Step 1: Draw the line of equation 1
Step 2: Draw the line of equation 2
Step 3: Use the point of intersection as the answers
3) Example
X + Y = 10 --------- Equation 1
2X – 3Y = 5 --------- Equation 2
Plot the table for X + Y = 10
X 2 4 6 8
Y 8 6 4 2
Plot the table for 2X – 3Y = 5
X 2 4 6 8
Y -0.33 1 2.33 3.67
Lastly, plot out the graph according to the tables.
Thus, the value of X and Y is 7 and 3 respectively.
4) Accuracy, Efficiency and Convenience
Accuracy- it is not accurate at all as a slight mistake might result in the change of answer like the lines are crooked or the plotting of the graph is wrong. Also, when dealing with fractions or decimals, there is a very high chance that we might divide the boxes unequally thus result in loss of marks. We might make human errors or estimate wrongly. Thus in terms of accuracy, it is quite unreliable.
Efficiency- it is not efficient, as we might need to redo or recalculate the tables of values if the lines do not cross each other. Also, there is a limitation to this method since we could not divide the boxes equally when there is fractions or decimals. The time taken to plot and draw the graph are longer than solving the questions using the other two methods.
Convenience- it is not convenience at all since we need to calculate numerous value before plotting and we could not use this method outside easily since when in need, we often do not carry calculators and graph paper along. The steps taken to find the values are also quite tedious and if we press wrongly, we will need to redo everything again and there is no time for that.
Jodi Chiam (9) 2A=D
Okay... this is my third time doing this maths journal...=_=
There are three different methods to solve simultaneous equations:they are the elimination method, the substitution method and the graphical solution method.
ELIMINATION METHOD
In this method, we first eliminate one of the two unknowns. Either one of the unknowns can be eliminated first. Pick the one that is easier to eliminate.
I prefer the elimination method as it saves the most time than using other method. Because it is the most straightforward and the least complicated one.We can easily arrive to the answer by mental calculation or by using the calculator. Therefore it makes it hard for us to make careless mistakes and we do have to redo it again.
SUBSTITUTION METHOD
In this method, we solve one of the equations for one unknown and substitute this expression into the other equation to solve for the remaining unknown.
Even though the steps that we need to take before solving the equation are shorter that using the elimination method and the graphical method, i am not that familiar with this method.Therefore i tend to make careless mistakes while using this method. And I find the steps complicated and confusing. Also i have to waste time checking for careless mistakes by redoing the question again.
GRAPHICAL METHOD
In this method, we have to draw the graphs of both equations on the same axes. Then read off the values of the variables at the point of intersection of the lines. The coordinates of the point of intersection gives the solution of the simultaneous equations.
This method takes up the most time. We have to waste time drawing the graph. And if we draw it wrongly, we have to redo the whole graph all over again. And when we read the graph, we may not be accurate causing an error in calculation and a wrong answer.
In conclusion, i prefer the elimination method=]
There are 3 different kinds of ways to solve the linear equations.
1. elimination method
2. substitution method
3. graphical method
I prefer to use the substitution method as its is the first method taught. (or is it?) anyway,
Elimination is when you eliminate one of the two unknowns. Either of it can be eliminated first. Choose the one that is easier to eliminate.
Substitution is when you rearrange the equation such that either an unknown is the subject of an equation. Than substitute this equation into the second equation and we had got the solution for one unknown. Than we can substitute the solution into the equation, getting the solution for the other unknown.
Graphical is when you plot a graph. If a solution exist in the equations, it will appear as the coordinates of the graphs of the simultaneous equations.But you do not always get an exact solution.
-Jingsi
ps..i rush through this as the library is closing soon. Also, i dont know what else to write. -.-
Post a Comment