Question

How do solve the following linear system?: `-6x+6y=6 , -6x+3y=-12`?

Answer 1

`(x,y) = (5,6)`

Explanation:

`6y=6x+6`
`y=x+1` solve for `y` in terms of `x` in the first equation
`-6x+3(x+1)=-12` plug into the other equation
`-6x+3x+3=-12` distribute
`3-3x=-12` combine like terms
`-3x=-15` subtract 3 from both sides
`x=5` divide both sides by -3

Now, going back to the other equation
`-6(5)+6y=6` plug in the value 5 for x
`-30+6y=6` multiply
`6y=36` subtract 30 from both sides
`y=6` divide both sides by 6

Answer 2

Subtract equation 2 from equation 1 to solve for `y`, and then back-solve for `x` to find that `x=5` and `y=6`

Explanation:

Let's use linear algebra to solve the system of equations. This will involve subtracting whole equations from each other:

`-6x+6y=6`
`ul(-(-6x+3y=-12))`

`-6x+6y=6`
`ul(+6x-3y=12)`
`cancel(0x)+3y=18`

Now that `x` is eliminated, we can solve for `y`:

`3y=18`

`rArr (cancel(3)y)/cancel(color(red)(3))=18/color(red)(3)`

`rArr color(blue)(y=6)`

Finally, we can plug our solution for `y` back into either equation to solve for `x`:

`-6x+3y=-12`

`rArr -6x+3(color(blue)(6))=-12`

`rArr -6x+18=-12`

`rArr -6xcancel(+18color(red)(-18))=-12color(red)(-18)`

`rArr -6x=-30`

`rArr (cancel(-6)x)/cancel(color(red)(-6))=(-30)/color(red)(-6)`

`rArr color(green)(x=5)`