College Algebra, Chapter 7, 7.3, Section 7.3, Problem 32

Solve the system of equations $\left\{
\begin{array}{cccccc}
x & +2y & & +3w & = & 0 \\
& y & + z & +w & = & 1 \\
& y & & + w & = & 2 \\
x & + 2y & & + 2w & = & 3
\end{array}
\right.
$, $\displaystyle \left[ \begin{array}{cccc}
0 & 0 & -2 & -1 \\
-1 & 0 & 1 & 1 \\
0 & 1 & -1 & 0 \\
1 & 0 & 0 & -1
\end{array} \right] $ by converting to a matrix equation and using the inverse of the coefficient matrix $\left[ \begin{array}{cc}
-9 & 4 \\
7 & -3
\end{array} \right]$

The equivalent matrix equation is







Using the formula for solving a matrix equation

$X = A^{-1} B$

We have







Thus, $x = -7, y = 5, z = -1$ and $w = -3$ is the solution of the original system.