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1. \begin{cases}x + y + z = 6 \\ x - y = 1 \\ y + z = 5\end{cases}

   The system has 3 equations in 3 unknowns (x, y, z).

2. \left[\begin{matrix}1 & 1 & 1 & 6\\1 & -1 & 0 & 1\\0 & 1 & 1 & 5\end{matrix}\right]

   Write the augmented matrix [A | b] and row-reduce it (Gaussian elimination).

3. \left[\begin{matrix}1 & 1 & 1 & 6\\0 & -2 & -1 & -5\\0 & 1 & 1 & 5\end{matrix}\right]

   R2 ← R2 − (1)·R1 to clear column 1.

4. \left[\begin{matrix}1 & 1 & 1 & 6\\0 & 1 & \frac{1}{2} & \frac{5}{2}\\0 & 1 & 1 & 5\end{matrix}\right]

   R2 ← R2 / -2 to make the pivot 1.

5. \left[\begin{matrix}1 & 0 & \frac{1}{2} & \frac{7}{2}\\0 & 1 & \frac{1}{2} & \frac{5}{2}\\0 & 1 & 1 & 5\end{matrix}\right]

   R1 ← R1 − (1)·R2 to clear column 2.

6. \left[\begin{matrix}1 & 0 & \frac{1}{2} & \frac{7}{2}\\0 & 1 & \frac{1}{2} & \frac{5}{2}\\0 & 0 & \frac{1}{2} & \frac{5}{2}\end{matrix}\right]

   R3 ← R3 − (1)·R2 to clear column 2.

7. \left[\begin{matrix}1 & 0 & \frac{1}{2} & \frac{7}{2}\\0 & 1 & \frac{1}{2} & \frac{5}{2}\\0 & 0 & 1 & 5\end{matrix}\right]

   R3 ← R3 / \frac{1}{2} to make the pivot 1.

8. \left[\begin{matrix}1 & 0 & 0 & 1\\0 & 1 & \frac{1}{2} & \frac{5}{2}\\0 & 0 & 1 & 5\end{matrix}\right]

   R1 ← R1 − (\frac{1}{2})·R3 to clear column 3.

9. \left[\begin{matrix}1 & 0 & 0 & 1\\0 & 1 & 0 & 0\\0 & 0 & 1 & 5\end{matrix}\right]

   R2 ← R2 − (\frac{1}{2})·R3 to clear column 3.

10. x = 1,\; y = 0,\; z = 5

    Read the solution off the reduced matrix.