`x - 3y + 2z = 18, 5x - 13y + 12z = 80` Solve the system of linear equations and check any solutions algebraically.

You should notice that the system has a smaller number of equations than the number of variables, hence, the system is indeterminate.

`x = 3y - 2z + 18`

Replace `3y - 2z + 18` for x in the second equation, such that:

`5(3y - 2z + 18) - 13y + 12z = 80 => 15y - 10z + 90 - 13y + 12z = 80`

`2y + 2z = -10 => y + z = -5 => y = -5-z => x = 3(-5-z) - 2z + 18`

`x = 3 - 5z`

Hence, evaluating the solutions to the system yields `x = 3 - 5z, y = -5-z, z = z.`