Question

How do you solve `1/2x+y=5` and `-x-2y=4` using matrices?

Answer 1

There are no solutions; the equations represent two parallel lines which do not intersect.

Explanation:

If you multiply the first equation by `(-2)` you will see that the new version of the first equation and the second equation have identical expressions on the left side but different values on the right side.

I you try to handle this as a matrix problem:

`((1/2,1,5),(-1,-2,4))`

using Cramer's Rule (determinants)
`x=|D_x|/|D|color(white)("XXX")y=|D_y|/|D|`

`|D| = |(1/2,1),(-1,-2)| = 1/2xx(-2)-(-1)xx2=0`

but division by `0` is not valid, so `x` and `y` can not be evaluated.

using Gauss-Jorden Method
`((1/2,1,5),(-1,-2,4))`

`rarr ((1,2,10),(-1,-2,-4))`

`rarr ((1,2,10),(0,0,6))`

both the `x` and `y` columns are zero, so the second row of this matrix implies:
`color(white)("XXX")0*x+0*y=6`
which is clearly impossible.