Questions: Solve the system. x+7y-z=49 2x-5y+2z=-28 x-6y=-32

Solution Steps

To solve the given system of linear equations, we can use matrix methods such as Gaussian elimination or matrix inversion. However, a straightforward approach is to use Python's numpy library, which provides a convenient function to solve systems of linear equations directly.

We are given the following system of equations: \[ \begin{align*}

1. & \quad x + 7y - z = 49 \\
2. & \quad 2x - 5y + 2z = -28 \\
3. & \quad x - 6y = -32 \end{align*} \]

This system can be expressed in matrix form as \( A \mathbf{x} = B \), where: \[ A = \begin{bmatrix} 1 & 7 & -1 \\ 2 & -5 & 2 \\ 1 & -6 & 0 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 49 \\ -28 \\ -32 \end{bmatrix} \]

By solving the matrix equation, we find: \[ \mathbf{x} = \begin{bmatrix} 4 \\ 6 \\ -3 \end{bmatrix} \] This gives us the values: \[ x = 4, \quad y = 6, \quad z = -3 \]

Final Answer