Write the following system of equations as a single matrix equation.
\[
\left\{\begin{aligned}
4 x-3 y+7 z & =-7 \\
3 x+2 y & =-14 \\
3 y-6 z & =3
\end{aligned}\right.
\]
\[\boxed{\begin{bmatrix} 4 & -3 & 7 \\ 3 & 2 & 0 \\ 0 & 3 & -6 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -7 \\ -14 \\ 3 \end{bmatrix}}\]
Step 1 :The given system of equations is
Step 2 :\[\begin{aligned} 4x - 3y + 7z &= -7 \\ 3x + 2y &= -14 \\ 3y - 6z &= 3 \end{aligned}\]
Step 3 :This system of equations can be written as a matrix equation of the form AX = B, where A is the matrix of coefficients, X is the column matrix of variables, and B is the column matrix of constants on the right side of the equations.
Step 4 :The matrix A is
Step 5 :\[\begin{bmatrix} 4 & -3 & 7 \\ 3 & 2 & 0 \\ 0 & 3 & -6 \end{bmatrix}\]
Step 6 :The matrix B is
Step 7 :\[\begin{bmatrix} -7 \\ -14 \\ 3 \end{bmatrix}\]
Step 8 :Therefore, the given system of equations can be written as a single matrix equation as follows
Step 9 :\[\boxed{\begin{bmatrix} 4 & -3 & 7 \\ 3 & 2 & 0 \\ 0 & 3 & -6 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -7 \\ -14 \\ 3 \end{bmatrix}}\]