\( \left[ \begin{array} { c c c | c } { 3 } & { - 2 } & { 4 } & { - 16 } \\ { 2 } & { 3 } & { - 2 } & { - 5 } \\ { 5 } & { - 1 } & { - 1 } & { - 3 } \end{array} \right] \)

To solve the system of linear equations represented by the augmented matrix: \[ \left[ \begin{array} { c c c | c } 3 & -2 & 4 & -16 \\ 2 & 3 & -2 & -5 \\ 5 & -1 & -1 & -3 \end{array} \right] \] we can use **Gaussian Elimination** to reduce the matrix to row-echelon form and then solve for the variables \( x \), \( y \), and \( z \). ### Step 1: Write the System of Equations The augmented matrix corresponds to the following system: \[ \begin{cases} 3x - 2y + 4z = -16 \quad \text{(Equation 1)} \\ 2x + 3y - 2z = -5 \quad \text{(Equation 2)} \\ 5x - y - z = -3 \quad \text{(Equation 3)} \end{cases} \] ### Step 2: Eliminate \( x \) from Equations 2 and 3 **a. Eliminate \( x \) from Equation 2** To eliminate \( x \) from Equation 2 using Equation 1: - Multiply Equation 1 by \( \frac{2}{3} \): \[ 2x - \frac{4}{3}y + \frac{8}{3}z = -\frac{32}{3} \] - Subtract this result from Equation 2: \[ \begin{align*} (2x + 3y - 2z) - \left(2x - \frac{4}{3}y + \frac{8}{3}z\right) &= -5 - \left(-\frac{32}{3}\right) \\ 0x + \left(3 + \frac{4}{3}\right)y + \left(-2 - \frac{8}{3}\right)z &= \frac{17}{3} \end{align*} \] Simplifying: \[ \frac{13}{3}y - \frac{14}{3}z = \frac{17}{3} \quad \text{(Equation 2')} \] Multiply through by 3 to eliminate fractions: \[ 13y - 14z = 17 \quad \text{(Equation 2')} \] **b. Eliminate \( x \) from Equation 3** To eliminate \( x \) from Equation 3 using Equation 1: - Multiply Equation 1 by \( \frac{5}{3} \): \[ 5x - \frac{10}{3}y + \frac{20}{3}z = -\frac{80}{3} \] - Subtract this result from Equation 3: \[ \begin{align*} (5x - y - z) - \left(5x - \frac{10}{3}y + \frac{20}{3}z\right) &= -3 - \left(-\frac{80}{3}\right) \\ 0x + \left(-1 + \frac{10}{3}\right)y + \left(-1 - \frac{20}{3}\right)z &= \frac{71}{3} \end{align*} \] Simplifying: \[ \frac{7}{3}y - \frac{23}{3}z = \frac{71}{3} \quad \text{(Equation 3')} \] Multiply through by 3 to eliminate fractions: \[ 7y - 23z = 71 \quad \text{(Equation 3')} \] ### Step 3: Solve for \( y \) and \( z \) Now, we have two equations with two variables: \[ \begin{cases} 13y - 14z = 17 \quad \text{(Equation 2')} \\ 7y - 23z = 71 \quad \text{(Equation 3')} \end{cases} \] **a. Eliminate \( y \)** - Multiply Equation 2' by 7: \[ 91y - 98z = 119 \quad \text{(Equation 2'')} \] - Multiply Equation 3' by 13: \[ 91y - 299z = 923 \quad \text{(Equation 3'')} \] - Subtract Equation 2'' from Equation 3'': \[ (91y - 299z) - (91y - 98z) = 923 - 119 \\ -201z = 804 \\ z = -4 \] **b. Solve for \( y \)** Plug \( z = -4 \) into Equation 2': \[ 13y - 14(-4) = 17 \\ 13y + 56 = 17 \\ 13y = -39 \\ y = -3 \] ### Step 4: Solve for \( x \) Use Equation 1 to solve for \( x \): \[ 3x - 2y + 4z = -16 \\ 3x - 2(-3) + 4(-4) = -16 \\ 3x + 6 - 16 = -16 \\ 3x - 10 = -16 \\ 3x = -6 \\ x = -2 \] ### Step 5: Summary of Solutions \[ \begin{cases} x = -2 \\ y = -3 \\ z = -4 \end{cases} \] **Therefore, the solution to the system of equations is:** \[ x = -2,\quad y = -3,\quad z = -4 \]