Questions: 6x + 5y + 2z = -1 -x + 3y + 7z = 12 5x - 7y - 3z = -52 (4,5,1) (5,-4,-1) (-1,5,-4) (-4,5,-1)

Transcript text: \[ \begin{array}{r} 6 x+5 y+2 z=-1 \\ -x+3 y+7 z=12 \\ 5 x-7 y-3 z=-52 \end{array} \] $(4,5,1)$ $(5,-4,-1)$ $(-1,5,-4)$ $(-4,5,-1)$

Solution

Solution Steps

To solve the given system of linear equations, we can use a method such as substitution, elimination, or matrix operations. Here, we'll use matrix operations to find the solution. We'll represent the system as a matrix equation $AX = B$, where $A$ is the coefficient matrix, $X$ is the column matrix of variables, and $B$ is the column matrix of constants. We'll then use Python's NumPy library to solve for $X$.

Step 1: Set Up the System of Equations

We are given the following system of linear equations:

\[ \begin{align_} 6x + 5y + 2z &= -1 \quad (1) \\ -x + 3y + 7z &= 12 \quad (2) \\ 5x - 7y - 3z &= -52 \quad (3) \end{align_} \]

Step 2: Represent the System in Matrix Form

We can represent the system in matrix form as $AX = B$, where:

\[ A = \begin{bmatrix} 6 & 5 & 2 \\ -1 & 3 & 7 \\ 5 & -7 & -3 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} -1 \\ 12 \\ -52 \end{bmatrix} \]

Step 3: Solve for the Variables

By solving the matrix equation $AX = B$, we find the values of $x$, $y$, and $z$:

\[ X = \begin{bmatrix} -4 \\ 5 \\ -1 \end{bmatrix} \]

This gives us:

\[ x = -4, \quad y = 5, \quad z = -1 \]