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How do I do this Simultaneous Equation

DabSchool

3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down

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Reply 1

B_9710

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.

Reply 2

razzor

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?

Reply 3

#engineer

Original post by DabSchool

3x+4y = 24
4x+3y = 22

Thanks, Would be helpful if you can break it down

Reply 4

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= 4.23
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

Reply 5

Ezme39

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 :smile:

very rough as it's without a calculator... seems close though looking at other peoples answers

Spoiler

(edited 7 years ago)

Reply 6

DabSchool

Original post by #engineer

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

Reply 7

RDKGames

Study Forum Helper

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.

Reply 8

RDKGames

Study Forum Helper

Original post by Ezme39

Haven't done these in years, so having a go for fun :smile:

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. :smile:

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.

(edited 7 years ago)

Reply 9

Ezme39

Original post by RDKGames

Good attempt, but you rounded your answers so your solutions are technically wrong.

Yeah, I just didn't have a calculator on me, so did rough mental maths instead :smile:

Reply 10

B_9710

Original post by Ezme39

Yeah, I just didn't have a calculator on me, so did rough mental maths instead :smile:

It's easier to give an answer as 30/7 without a calculator than as a decimal, no?

Reply 11

Ezme39

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 :smile:

but with a calculator of course you could just keep the number in there, and proceed with the next calculation

Reply 12

B_9710

Original post by Ezme39

Not to then put it in the equation, imo :smile:

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?

Reply 13

Ezme39

Original post by B_9710

Even so it's surely easier to multiply fractions than decimals?

No multiplying was needed, just subtracting :smile:

I wouldn't try to multiply decimals or fractions without paper :biggrin:

Reply 14

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.

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.

(edited 7 years ago)

Reply 15

Notnek

Original post by alephu5

Very detailed explanation but I feel you may be over-complicating things for a set of simple 2 variable linear simultaneous equations.

(edited 7 years ago)

Reply 16

alephu5

The OP asked

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.