Question

How do you solve the following system?: `16x +3y =7, 9x +5y = -12`

Answer 1

`(x,y)=(71/53, -255/53)`

Explanation:

Given
[1]`color(white)("XXX")16x+3y=7`
[2]`color(white)("XXX")9x+5y=-12`

Multiply [1] by `5` and [2] by `3` to get equations with equal coefficients for `y`
[3]`color(white)("XXX")80x+15y=35`
[4]`color(white)("XXX")27x+15y=-36`

Subtract [4] from [3]
[5]`color(white)("XXX")53x=71`

Divide by `23`
[6]`color(white)("XXX")x=71/53`
`color(white)("XXX")`Yes; it's going to be ugly; but that doesn't mean it's going to be wrong.

Substitute `71/53` for `x` in [1]
[7]`color(white)("XXX")(16xx71)/53+3y=7`

Simplify
[8]`color(white)("XXX")1136/53+159/53y=371/53`

[9]`color(white)("XXX")159y=-765`

[10]`color(white)("XXX")y=-765/159=-255/23`

I would (strongly) recommend using a calculator or spreadsheet to verify these values in the original equations.