Question

How do you solve the following system?: `-8x -7y =7, 5x -3y = -4`

Answer 1

`(x,y)=(-49/59,-3/59)`

Explanation:

Using matrices and Cramer's Rule
(with a bit of practice this can be done mentally, but I will lay out the steps in detail)

Converting
`color(white)("XXX")-8x-7y=7`
and
`color(white)("XXX")5x-3y=-4`
into matrix form:
`color(white)("XXXXX")xcolor(white)("XX")ycolor(white)("XX)c`
`color(white)("XXX")((-8,-7,7),(5,-3,-4))`
Using the determinants
`color(white)("XXX")D_c` (the original matrix without the `c` column)
`color(white)("XXXXX")=|(-8,-7),(5,-3)|=(-8) * (-3)-(-7) * (5) = 59`
Similarly
`color(white)("XXX")D_x=|(7,-7),(-4,-3)|=7 * (-3) - (-7) * (-4)=-49`
and
`color(white)("XXX")D_y=|(-8,7),(5,-4)|= (-8) * (-4) -5 * 7 = -3`

By Cramer's Rule
`color(white)("XXX")x=(D_x)/(D_c) = -49/59`
and
`color(white)("XXX")y=(D_y)/(D_c) = -3/59`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Alternately: By Elimination
[1]`color(white)("XXX")-8x-7y=7`
[2]`color(white)("XXX")5x-3y=-4`

[3]=[1]`xx 5` `color(white)("XXX")-40x-35y=35`
[4]=[2]`xx8` `color(white)("XXX")40x-24y=-32`

[5]=[3]+[4]`color(white)("XXX")-59y=3`
[6]`color(white)("XXXXXXXXX")y=-3/59`

...and similarly for `x`