You are Here: Home >< Maths

# How do I do this Simultaneous Equation

Announcements Posted on
TSR's new app is coming! Sign up here to try it first >> 17-10-2016
1. 3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down
2. (Original post by DabSchool)
3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down
Multiply first equation by 4 and second equation by 3 to get coefficients of x the same (could be done for y if you wanted) and then solve as you would if the coefficients were the same.
3. (Original post by DabSchool)
3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down
First, make sure the number in front of the x (or the y) are the same in both equations.
What could we multiply the first and second equation by so that the number in front of the x is the same?
4. (Original post by DabSchool)
3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down
5. the x's and the y's must equal each other out. to make it easier do first the x's and then the y's. therefore you need to multiply the upper equation with 4 and the lower equation with -3 to make the x's equal to 0 therefore:
4(3x+4y=24)-3(4x+3y=22)you need to multiply both equations therefore : 12x+16y= 96
-12x-9y=-66now : 12x-12x=0 therefore you solve for y :16y-9y=7y and 96-66= 30 your equation should look like this : 0x + 7y = 30 7y=30 y= 4.23
And then you solve for x (by putting where y is 4.23) : 12x + 16(4.23) = 96
12x + 67.68 = 96
12x = 28.32
x= 2.36
6. (Original post by DabSchool)
3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down
Haven't done these in years, so having a go for fun very rough as it's without a calculator... seems close though looking at other peoples answers
Spoiler:
Show
Equal the X or Y; I've chosen X
12x + 16y = 9612x + 9y = 66
Put the Xs to one side
12x = 96 - 16y
12x = 66 - 9y
Put the equations together
96 - 16y = 66 - 9y
Work it through to find Y
96 = 66 + 7y
7y = 30y = 4.3
Put Y into the original equation
3x + 17.2 = 24
3x = 6.8
x = 2.3
7. Thank you (and everyone else). Few questions...

1) Can you work with X and Y to get it the same ?

2) Do you have to write 30 over 7 and why
8. (Original post by DabSchool)
Thank you (and everyone else). Few questions...

1) Can you work with X and Y to get it the same ?

2) Do you have to write 30 over 7 and why
1) What do you mean?
2) Yes because it's an exact solution. Better get comfortable with fractions fast.
9. (Original post by Ezme39)
Haven't done these in years, so having a go for fun very rough as it's without a calculator... seems close though looking at other peoples answers

Equal the X or Y; I've chosen X
12x + 16y = 96
12x + 9y = 66
Put the Xs to one side
12x = 96 - 16y
12x = 66 - 9y
Put the equations together
96 - 16y = 66 - 9y
Work it through to find Y
96 = 66 + 7y
7y = 30
y = 4.3
Put Y into the original equation
3x + 17.2 = 24
3x = 6.8
x = 2.3
Good attempt, but you rounded your answers so your solutions are technically wrong. Also let's not post full solutions on this forum, thanks.

(Original post by Evita Tzn)
the x's and the y's must equal each other out. to make it easier do first the x's and then the y's. therefore you need to multiply the upper equation with 4 and the lower equation with -3 to make the x's equal to 0 therefore:4(3x+4y=24)-3(4x+3y=22)you need to multiply both equations therefore : 12x+16y= 96-12x-9y=-66now : 12x-12x=0 therefore you solve for y :16y-9y=7y and 96-66= 30 your equation should look like this : 0x + 7y = 30 7y=30 y= 4.23And then you solve for x (by putting where y is 4.23) : 12x + 16(4.23) = 9612x + 67.68 = 9612x = 28.32x= 2.36
Same for you.
10. (Original post by RDKGames)
Yeah, I just didn't have a calculator on me, so did rough mental maths instead
11. (Original post by Ezme39)
Yeah, I just didn't have a calculator on me, so did rough mental maths instead
It's easier to give an answer as 30/7 without a calculator than as a decimal, no?
12. (Original post by B_9710)
It's easier to give an answer as 30/7 without a calculator than as a decimal, no?
Not to then put it in the equation, imo but with a calculator of course you could just keep the number in there, and proceed with the next calculation
13. (Original post by Ezme39)
Not to then put it in the equation, imo but with a calculator of course you could just keep the number in there, and proceed with the next calculation
Even so it's surely easier to multiply fractions than decimals?
14. (Original post by B_9710)
Even so it's surely easier to multiply fractions than decimals?
No multiplying was needed, just subtracting I wouldn't try to multiply decimals or fractions without paper
15. Whenever you see a simultaneous equation you should think of overlapping curves or surfaces. If your equation has two variables (x, y) then it is a curve, if it has three variables it is a surface, if it has n it is an (n-1) - dimensional manifold. Why the -1? Well that's because one of the variables is determined by the equation, so in the case of a line x + 2y = 0, a given x completely determines y.

It is important to notice that I can multiply this equation by any number to produce the same line, e.g. x + 2y = 2x + 4y = 10x + 20y = 0 etc. Any other linear equation will produce a different line.

If you just work with just one of the equations you will never find numerical values for x or y unless it is a horizontal or vertical line. That is because there are an infinite number of solutions (x,y) to a line.

If you are hoping to twiddle 3x + 4y = 24 around to get 4x + 3y = blah, I'm afraid that cannot ever happen. The first equation should be thought of as an assumption/assertion that "this line exists.". It should be obvious intuitively that the second equation, equivalent to the assertion that "this other line also exists.", does not follow logically. A single line existing by itself is perfectly fine, so the existence of a second equation is an assumption plucked out of thin air. You could choose to have more lines if you wanted.

Algebraic twiddling is equivalent to a pure logical argument; which is to say it does not change the properties of the line e.g. 3x + 4y = 24 is the same line as -3x - 4y = -24.

What is special is that we are using (x,y) in both equations, so implicitly we have the added assumption that "there exists a point (x,y) on both lines." Such a point does not always exist; for example if the second equation were 3x + 4y = 25 the second line would be parallel and offset. If there are more than 2 equations of a line in 2D space, it might also fail.

To attempt to find a solution, you simply need to combine the two (or more) equations into a single equation. It really does not matter how you do this, provided you don't stray from the laws of algebra. A common way is to manipulate the variables in both equations independently until you get A = blah and B = blah, then take A = B and solve for x or y. This may not be numerical, but if you substitute your solution for into either of the equations you are certain to obtain a numerical solution.

The problem of solving simultaneous linear equations is extremely important in every technical field and it is right at the heart of quantum mechanics. We call it linear algebra and usually use matrix representations, which are grids of numbers or variables. With this approach you can solve such equations very easily, although the algorithm seems weird until you get a bit deeper into the maths.
16. (Original post by alephu5)
Whenever you see a simultaneous equation you should think of overlapping curves or surfaces. If your equation has two variables (x, y) then it is a curve, if it has three variables it is a surface, if it has n it is an (n-1) - dimensional manifold. Why the -1? Well that's because one of the variables is determined by the equation, so in the case of a line x + 2y = 0, a given x completely determines y...
Very detailed explanation but I feel you may be over-complicating things for a set of simple 2 variable linear simultaneous equations.

1) Can you work with X and Y to get it the same ?
which suggested a fundamental lack of understanding in what a simultaneous equation represents. I wanted to exhibit some of the intuition behind them in an attempt to help.
18. (Original post by #engineer)
Is this Gcse maths Grade C question?
19. (Original post by ubi1)
Is this Gcse maths Grade C question?
In the old GCSE this would be a grade B question. In the new GCSE it is a grade 5 question and is part of foundation.
20. Not very clear but I've attached a guide to solving this equation:

HTH

## Register

Thanks for posting! You just need to create an account in order to submit the post
1. this can't be left blank
2. this can't be left blank
3. this can't be left blank

6 characters or longer with both numbers and letters is safer

4. this can't be left empty
1. Oops, you need to agree to our Ts&Cs to register

Updated: September 8, 2016
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Today on TSR

### How does exam reform affect you?

From GCSE to A level, it's all changing

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read here first

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams