Solve $$$\begin{cases} a + b + c = 4 \\ - b + c = 0 \\ - a = -2 \end{cases}$$$ for $$$a$$$, $$$b$$$, $$$c$$$

Solve $$$\begin{cases} a + b + c = 4 \\ - b + c = 0 \\ - a = -2 \end{cases}$$$ for $$$a$$$, $$$b$$$, $$$c$$$ using the Gauss-Jordan Elimination method.

Write down the augmented matrix: $$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 4\\0 & -1 & 1 & 0\\-1 & 0 & 0 & -2\end{array}\right]$$$.

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 4\\0 & -1 & 1 & 0\\0 & 0 & 2 & 2\end{array}\right]$$$.

Back-substitute:

$$$c = \frac{2}{2} = 1$$$

$$$b = \frac{0 - \left(1\right)^{2}}{-1} = 1$$$

$$$a = 4 - \left(1\right)^{2} - \left(1\right)^{2} = 2$$$

$$$a = 2$$$A

$$$b = 1$$$A

$$$c = 1$$$A