# How do you solve -x + y + z = -1 and -x + 5y - 15z = -13 and 3x - 2y - 7z = 0 using matrices?

Feb 22, 2016

$\Delta = \left[\begin{matrix}- 1 & 1 & 1 \\ - 1 & 5 & - 15 \\ 3 & - 2 & - 7\end{matrix}\right] = 35 - 45 + 2 - \left(15 - 30 + 7\right) = - 8 + 8 = 0$
$\Delta x = \left[\begin{matrix}- 1 & 1 & 1 \\ - 13 & 5 & - 15 \\ 0 & - 2 & - 7\end{matrix}\right] = 35 + 0 + 26 - \left(0 - 30 + 91\right) = 61 - 61 = 0$
$\Delta y = \left[\begin{matrix}- 1 & - 1 & 1 \\ - 1 & - 13 & - 15 \\ 3 & 0 & - 7\end{matrix}\right] = - 91 + 45 + 0 - \left(- 39 - 7 + 0\right) = - 46 + 46 = 0$
$\Delta z = \left[\begin{matrix}- 1 & 1 & - 1 \\ - 1 & 5 & - 13 \\ 3 & - 2 & 0\end{matrix}\right] = 0 - 39 - 2 - \left(15 - 26 + 0\right) = - 41 + 11 = - 10$
Since $\Delta = 0$ but $\Delta z \ne 0$ this system of equations is a impossible one and has no solution.