Solve $$$\begin{cases} a + b = 0 \\ - a + b = 1 \end{cases}$$$ for $$$a$$$, $$$b$$$

Solve $$$\begin{cases} a + b = 0 \\ - a + b = 1 \end{cases}$$$ for $$$a$$$, $$$b$$$ using the Gauss-Jordan Elimination method.

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

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

Back-substitute:

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

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

$$$a = - \frac{1}{2} = -0.5$$$A

$$$b = \frac{1}{2} = 0.5$$$A