Write the following system of equations as a single matrix equation. \[ \left\{\begin{aligned} 4 x-3 y+7 z & =-7 \\ 3 x+2 y & =-14 \\ 3 y-6 z & =3 \end{aligned}\right. \]

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 ：\[\boxed{\begin{bmatrix} 4 & -3 & 7 \\ 3 & 2 & 0 \\ 0 & 3 & -6 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -7 \\ -14 \\ 3 \end{bmatrix}}\]