\( \left. \begin{array} { l l } { x - 3 y + z = 7 } & { x = } \\ { - 2 x + 3 y + 3 z = - 16 } & { y = } \\ { y = - 6 } & { z = \square } \end{array} \right. \)

To solve the system of equations, we can use the substitution method. Given equations: 1. \(x - 3y + z = 7\) 2. \(-2x + 3y + 3z = -16\) 3. \(y = -6\) Substitute the value of \(y\) from equation 3 into equations 1 and 2: 1. \(x - 3(-6) + z = 7\) 2. \(-2x + 3(-6) + 3z = -16\) Simplify the equations: 1. \(x + 18 + z = 7\) 2. \(-2x - 18 + 3z = -16\) Now, we have two equations with two variables \(x\) and \(z\). We can solve these equations to find the values of \(x\) and \(z\). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+18+z=7\\-2x-18+3z=-16\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-11-z\\-2x-18+3z=-16\end{array}\right.\) - step2: Substitute the value of \(x:\) \(-2\left(-11-z\right)-18+3z=-16\) - step3: Simplify: \(4+5z=-16\) - step4: Move the constant to the right side: \(5z=-16-4\) - step5: Subtract the numbers: \(5z=-20\) - step6: Divide both sides: \(\frac{5z}{5}=\frac{-20}{5}\) - step7: Divide the numbers: \(z=-4\) - step8: Substitute the value of \(z:\) \(x=-11-\left(-4\right)\) - step9: Simplify: \(x=-7\) - step10: Calculate: \(\left\{ \begin{array}{l}x=-7\\z=-4\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=-7\\z=-4\end{array}\right.\) - step12: Rewrite: \(\left(x,z\right) = \left(-7,-4\right)\) The solution to the system of equations is \(x = -7\) and \(z = -4\). Therefore, the value of \(z\) is \(-4\).