Question

How do you solve `x + 6y = 28` and `2x - 3y = -19`?

Answer 1

Double the second equation, and then add the two equations together linearly, solving for `x`, and then solve for `y` in either. You will find that `x=-2` and `y=5`

Explanation:

First, line up the equations with one above the other:

`x+6y=28`
`2x-3y=-19`

next, multiply the ENTIRE bottom equation by 2:

`x+6y=28`
`color(red)(4)x-color(red)(6)y=color(red)(-38)`

Now, add the equations together:

`(x+6y=28)`
`ul(+(4x-6y=-38)`
`color(red)((4+1))x+color(red)((6-6))y=color(red)((28-38))`

`color(red)(5)x+color(red)(0)y=color(red)(-10)`

`5x=-10`

Now, solve for `x`:

`(cancel(5)x)/color(red)(cancel(5))=(-10)/color(red)(5)`

`color(blue)(x=-2)`

Since we now have a solution for `x`, we can plug it back into either equation to solve for `y`. Doing it in both proves that our solution for `x` is valid:

Equation 1:

`color(blue)(x)+6y=28`

`color(blue)(-2)+6y=28`

`color(blue)(cancel(-2))+6ycolor(red)(cancel(+2))=28color(red)(+2)`

`6y=30`

`(cancel(6)y)/color(red)(cancel(6))=30/color(red)(6)`

`color(green)(y=5)`

Equation 2:

`2color(blue)(x)-3y=-19`

`2color(blue)((-2))-3y=-19`

`-4-3y=-19`

`cancel(-4)-3ycolor(red)(cancel(+4))=-19color(red)(+4)`

`-3y=-15`

`(cancel(-3)y)/color(red)(cancel(-3))=(-15)/color(red)(-3)`

`color(green)(y=5)`

Both equations support the solution, we're done!