Question

How do you solve `5x-y-7z= -13`, `-3x +8z=22`, `9y+z=77` using matrices?

Answer 1

One way would be to use Cramer's Rule, to get
`color(white)("XXX")(x,y,z)=(6,8,5)`

Explanation:

The given equations could be written in matrix form as:
`((x,y,z,"|",c),(5,-1,-7,"|",-13),(-3,0,8,"|",22),(0,9,1,"|",77))`
`color(white)("XXX")`actually the top row shouldn't be there, but I though it might make the translation more clear.

Cramer's Rule says that if you take the derivatives:
`color(white)("XXX")D =` the matrix composed of the variable coefficients (the left side of the above)
`color(white)("XXX")D_x =` the matrix composed of the variable coefficients with the `x` column replaced with the `c` column.
`color(white)("XXX")D_y =` the matrix composed of the variable coefficients with the `y` column replaced with the `c` column.
`color(white)("XXX")D_z =` the matrix composed of the variable coefficients with the `z` column replaced with the `c` column.

then
`color(white)("XXX")x=|D_x|/|D|color(white)("XXX")y=|D_y|/|D|color(white)("XXX")z=|D_z|/|D|`

Here's what the solution looks like done on a spreadsheet: