Question

How do you solve `2x + 7y = - 8` and `- 2x + 3y = - 12` using matrices?

Answer 1

`(x,y)=(3,-2)`
(see below for use of matrices)

Explanation:

Define the matrices:
#M_(xyc)=({: (x,y,=c), (2,7,-8), (-2,3,-12) :})#

`M_(xy)=({:(2,7),(-2,3):})color(white)("XX")M_(cy)=({:(-8,7),(-12,3):})color(white)("XX")M_(xc)=({:(2,-8),(-2,-12):})`

and their determinants:
`D_(xy)=|{:(2,7),(-2,3):}|=2xx3-(-2)xx7=20`

`D_(cy)=|{:(-8,7),(-12,3):}|=(-8)xx3-(-12)xx7=60`

`D_(xc)=|{:(2,-8),(-2,-12):}|=2xx(-12)-(-2)xx(-8)=-40`

By Cramer's Rule
`color(white)("XXX")x=(D_(cy))/(D_(xy))=60/20=3`
and
`color(white)("XXX")y=(D_(xc))/(C_(xy))=(-40)/20=-2`