1. We are given the augmented matrix for a system of equations: $$\begin{bmatrix}5 & 2 & | & 8 \\ 3 & 1 & | & 6\end{bmatrix}$$ 2. From the matrix, the system can be written as: $$ 5x + 2y = 8 $$ $$ 3x + y = 6 $$ 3. To solve, start with the second equation: $$ y = 6 - 3x $$ 4. Substitute this expression for $y$ into the first equation: $$ 5x + 2(6 - 3x) = 8 $$ 5. Distribute and simplify: $$ 5x + 12 - 6x = 8 $$ $$ (5x - 6x) + 12 = 8 $$ $$ -x + 12 = 8 $$ 6. Solve for $x$: $$ -x = 8 - 12 $$ $$ -x = -4 $$ $$ x = 4 $$ 7. Substitute $x=4$ back into $y = 6 - 3x$: $$ y = 6 - 3(4) = 6 - 12 = -6 $$ 8. The solution to the system is: $$ (x, y) = (4, -6) $$