Questions: Solve the system using a matrix. -x-7y-z=-19 4x+4y+4z=4 2x+y+6z=7 Give your answer as an ordered triple (x, y, z).

Transcript text: Solve the system using a matrix. \[ \begin{array}{c} \left\{\begin{array}{l} -x-7 y-z=-19 \\ 4 x+4 y+4 z=4 \\ 2 x+y+6 z=7 \end{array}\right. \\ \end{array} \] Give your answer as an ordered triple ( $x, y, z$ ).

Solution

Solution Steps

To solve the system of linear equations using a matrix, we can represent the system in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the column matrix of variables, and $B$ is the column matrix of constants. We then use numpy's linear algebra solver to find the solution.

Step 1: Represent the System of Equations in Matrix Form

We start by representing the given system of linear equations in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the column matrix of variables, and $B$ is the column matrix of constants.

\[ A = \begin{bmatrix} -1 & -7 & -1 \\ 4 & 4 & 4 \\ 2 & 1 & 6 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} -19 \\ 4 \\ 7 \end{bmatrix} \]

Step 2: Solve the Matrix Equation

To find the solution $X$, we solve the matrix equation $AX = B$. The solution is given by:

\[ X = A^{-1}B \]

Step 3: Compute the Solution

After performing the matrix operations, we find the solution for $X$:

\[ X = \begin{bmatrix} -4.000 \\ 3.000 \\ 2.000 \end{bmatrix} \]