Solve $$$\begin{cases} a + b + c = 4 \\ 4 a + 2 b + c = 9 \\ a - b + c = 6 \end{cases}$$$ for $$$a$$$, $$$b$$$, $$$c$$$

Solve $$$\begin{cases} a + b + c = 4 \\ 4 a + 2 b + c = 9 \\ a - b + c = 6 \end{cases}$$$ for $$$a$$$, $$$b$$$, $$$c$$$ using the Gauss-Jordan Elimination method.

Write down the augmented matrix: $$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 4\\4 & 2 & 1 & 9\\1 & -1 & 1 & 6\end{array}\right]$$$.

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{ccc|c}1 & 1 & 1 & 4\\0 & -2 & -3 & -7\\0 & 0 & 3 & 9\end{array}\right]$$$.

Back-substitute:

$$$c = \frac{9}{3} = 3$$$

$$$b = \frac{-7 - \left(-3\right) \left(3\right)}{-2} = -1$$$

$$$a = 4 - \left(1\right) \left(3\right) - \left(-1\right) \left(1\right) = 2$$$

$$$a = 2$$$A

$$$b = -1$$$A

$$$c = 3$$$A