Question

How do you solve `x - 2y = 5` and `2x - 3y = 10` using matrices?

Answer 1

Given
`color(white)("XXX") x-2y=5`
`color(white)("XXX") 2x-3y=10`

Re-writing as augmented matrix:
`color(white)("XXX") [(1,-2,5),(2,-3,10)]color(white)("XXX"){:([1]),([2]):}`

Subtracting `2` times the first row from the second row:
`color(white)("XXX") [(1,-2,5),(0,1,0)]color(white)("XXX"){:([1]),([3]=[2]-2xx[1]):}`

Adding `2` times row [3] to the first row:
`color(white)("XXX") [(1,0,5),(0,1,0)]color(white)("XXX"){:([4]),([3]=[1]+2xx[3]):}`

Re-writing with variables:
`color(white)("XXX")x=5`
`color(white)("XXX")y=0`