Rewrite the system of equations in matrix form.
[tex]\begin{bmatrix}1&2&-2\\3&7&-1\\2&4&m\end{bmatrix} \mathbf x = \mathbf b[/tex]
This system has a unique solution [tex]\mathbf x = \mathbf A^{-1}\mathbf b[/tex] so long as the inverse of the coefficient matrix [tex]\mathbf A[/tex] exists. This is the case if the determinant is not zero.
We have
[tex]\det(\mathbf A) = m+4[/tex]
so the inverse, and hence a unique solution to the system of equations, exists as long as m ≠ -4.
