Megoldani a matematikai problémákat

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1. \begin{cases}x + y + z = 6 \\ 2 x - y = 0 \\ x + 2 z = 7\end{cases}

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

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

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

3. \left[\begin{matrix}1 & 1 & 1 & 6\\0 & -3 & -2 & -12\\1 & 0 & 2 & 7\end{matrix}\right]

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

4. \left[\begin{matrix}1 & 1 & 1 & 6\\0 & -3 & -2 & -12\\0 & -1 & 1 & 1\end{matrix}\right]

   R3 ← R3 − (1)·R1 to clear column 1.

5. \left[\begin{matrix}1 & 1 & 1 & 6\\0 & 1 & \frac{2}{3} & 4\\0 & -1 & 1 & 1\end{matrix}\right]

   R2 ← R2 / -3 to make the pivot 1.

6. \left[\begin{matrix}1 & 0 & \frac{1}{3} & 2\\0 & 1 & \frac{2}{3} & 4\\0 & -1 & 1 & 1\end{matrix}\right]

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

7. \left[\begin{matrix}1 & 0 & \frac{1}{3} & 2\\0 & 1 & \frac{2}{3} & 4\\0 & 0 & \frac{5}{3} & 5\end{matrix}\right]

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

8. \left[\begin{matrix}1 & 0 & \frac{1}{3} & 2\\0 & 1 & \frac{2}{3} & 4\\0 & 0 & 1 & 3\end{matrix}\right]

   R3 ← R3 / \frac{5}{3} to make the pivot 1.

9. \left[\begin{matrix}1 & 0 & 0 & 1\\0 & 1 & \frac{2}{3} & 4\\0 & 0 & 1 & 3\end{matrix}\right]

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

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

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

11. x = 1,\; y = 2,\; z = 3

    Read the solution off the reduced matrix.