Question

How do you solve the following linear system: # -x+2y=-6 , 6x-y=-35 #?

Answer 1

`(-76/11 , -71/11)`

Explanation:

`-x+2y=-6`
`6x-y=-35`

Let's use linear combination to solve this system.

We will begin by multiplying the second system by `2`.

`12x-2y=-70`

Then, we will add the two equations together. This works because we add `-70` to one side and add `(12x-2y)`, an equivalent value, to the other side.

`11x=-76`
`x=-76/11`

Then we'll find, in either equation, the output `y` value for this value of `x`.

`-(color(red)(-76/11))+2y=-6`
`color(red)(76/11)+2y=-6`
`2y=color(red)(-142/11)`
`y=-71/11`

These systems intersect at the point `(-76/11 , -71/11)`

This solution is pretty ugly, so let's check to make sure it's correct.
We'll use `6x-y=-35`, and substitute `-71/11` for `y`

`6x-(color(red)(-71/11))=-35`
`6xcolor(red)(+71/11)=-35`
`6x=color(red)(-456/11)`
`x=-76/11`

So our solution is `(-76/11 , -71/11)`.