Question

How do you solve the system of equations `-x-y=15` and `-8x+8y=24`?

Answer 1

`{(x=-9), (y=-6) :}`

Explanation:

The system of equations given to you looks like this

`{( -color(white)(1)x - color(white)(1)y = 15), (-8x + 8y = 24) :}`

Notice that one equation features `y` with a positive sign and the other has it with a negative sign. This means that if you get the coefficients to match, you can add the two equations and get rid of the `y` terms.

To do that, multiply the first equation by `8`

`{( -color(white)(1)x - color(white)(1)y = 15" " | xx 8), (-8x + 8y = 24) :}`

this will get you

`{ ( -8x - 8y = 120), (-8x + 8y = color(white)(1)24) :}`

Now you're ready to add the two equations

`{ ( -8x - 8y = 120), (-8x + 8y = color(white)(1)24) :}`
`color(white)(aaaaaaaaaaaaaaa)/color(white)(a)`

`-8x + (-8x) - color(red)(cancel(color(black)(8y))) + color(red)(cancel(color(black)(8y))) = 120 + 24`

`-16x = 144 implies x = 144/(-16) = -9`

Take this value of `x` into the first equation and find the value of `y`

`- (-9) - y = 15`

`-y = 15 - 9 implies y = -6`

Therefore, the two solutions to your system of equations are

`{(x=-9), (y=-6) :}`