Question

How do you solve using gaussian elimination or gauss-jordan elimination, `4x-3y= -1`, `3x+4y= -3`?

Answer 1

`x=-0.52`
`y=-0.36`

Explanation:

Given
`color(white)("XXX")4x-3y=-1`
`color(white)("XXX")3x+4y=-3`

In augmented matrix form:
#[ (4, -3, "|", -1), (3, 4, "|",-3) ]#

Starting with row 1 as the "pivot row" (and column 1 as the "pivot column")
Reduce the pivot row (by dividing by the pivot entry value) so the pivot entry `=1`
#[ (4/4, -3/4, "|", -1/4), (3, 4, "|",-3) ]=[ (1, -0.75, "|", -0.25), (3, 4, "|",-3) ]#

Reduce the non-pivot rows so their pivot column entries are `0`
(by subtracting the pivot row times the non-pivot row, pivot column entry from the non-pivot row)
#[ (1, -0.75, "|", -0.25), (3-3xx1, 4-3xx(-0.75), "|",-3-3xx(-0.25)) ]=[ (1, -0.75, "|", -0.25), (0, 6.25, "|",-2.25) ]#

Advance the pivot row (so it is now row 2)
and repeat the above process:
#[ (1, -0.75, "|", -0.25), (0, 1, "|",-0.36) ]#

then
#[ (1, 0, "|", -0.52), (0, 1, "|",-0.36) ]#