Question

Let k, h be unknown constants and consider the linear system:

x - 4y + 7z = -7
7x - 4y +8z = 3
17x - 20y + hz = k
This system has a unique solution whenever h does not equal ???

Answer 1

If the determinant of the coefficients is zero, there is no unique solution.
|1 -4 7|
|7 -4 8|
|17 -20 h| = 24h-120
so as long as h≠5 there is a unique solution.