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

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Answer 1 · 2017-06-23T12:33:52

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

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
$color(white)("XXX")y=(D_(xc))/(C_(xy))=(-40)/20=-2$

How do you solve 2x + 7y = - 82x + 7y = - 8 and - 2x + 3y = - 12- 2x + 3y = - 12 using matrices?