Question

How do you solve `3x+3y=9, -4x+y=3`?

Answer 1

Use linear combination to get the solution `(0,3)`.

Explanation:

`color(white)(aa)3x+3y=9`
`color(red)(-4x+color(white)ay=3)`

Multiiply the 2nd equation by `-3`. This will allow you to "cancel out" the `y` terms by adding the equations together. This technique is usually called "linear combination" or sometimes "elimination".

`-3(color(red)(-4x+y=3))`

`color(red)(12x-3y=-9)`

Rewrite with the first equation and add the two equations together.

`color(white)(\a)3x+3y=color(white)(aa)9`
`color(red)(12x-3y=-9)`

`15x=0`

`(15x)/15=0/15color(white)(aaa)`Divide both sides by 15

`x=0`

Substitute `x=0` into either of the original equations. I will pick the first equation.

`3x+3y=9`

`3(0)+3y=9`

`3y=9`

`(3y)/3=9/3color(white)(aaa)`Divide both sides by 3

`y=3`

The solution is `(0,3)`.