Write the system of equations as an augmented matrix \( \left\{\begin{array}{l}-4 x+6 y-5 z=-55 \\ -3 x-7 y-z=-144 \\ 5 x-2 y+2 z=68\end{array}\right. \) \( \square \)

To write the given system of equations as an augmented matrix, we first identify the coefficients of the variables \(x\), \(y\), and \(z\) in each equation, as well as the constants on the right side of the equations. The system of equations is: \[ \begin{align*} -4x + 6y - 5z &= -55 \\ -3x - 7y - z &= -144 \\ 5x - 2y + 2z &= 68 \end{align*} \] Now, we can extract the coefficients and the constants: 1. For the first equation \(-4x + 6y - 5z = -55\), the coefficients are \(-4\), \(6\), and \(-5\), and the constant is \(-55\). 2. For the second equation \(-3x - 7y - z = -144\), the coefficients are \(-3\), \(-7\), and \(-1\), and the constant is \(-144\). 3. For the third equation \(5x - 2y + 2z = 68\), the coefficients are \(5\), \(-2\), and \(2\), and the constant is \(68\). Now we can construct the augmented matrix, which includes the coefficients of the variables and the constants in the last column: \[ \begin{bmatrix} -4 & 6 & -5 & | & -55 \\ -3 & -7 & -1 & | & -144 \\ 5 & -2 & 2 & | & 68 \end{bmatrix} \] In standard notation, the augmented matrix can be written as: \[ \begin{bmatrix} -4 & 6 & -5 & -55 \\ -3 & -7 & -1 & -144 \\ 5 & -2 & 2 & 68 \end{bmatrix} \] This is the augmented matrix corresponding to the given system of equations.