11. Solve the system of equations using matrices (Gaussian elimination). Record each row operation that you perform so that we can follow your work. (6 points) \[ \begin{array}{c}x+y+z=-4 \\ 2 x+y-z=7 \\ x+2 y+2 z=3\end{array} \]

Solve the system of equations \( x+y+z=-4;2x+y-z=7;x+2y+2z=3 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y+z=-4\\2x+y-z=7\\x+2y+2z=3\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-4-y-z\\2x+y-z=7\\x+2y+2z=3\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left\{ \begin{array}{l}2\left(-4-y-z\right)+y-z=7\\-4-y-z+2y+2z=3\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}-8-y-3z=7\\-4+y+z=3\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}y=-15-3z\\-4+y+z=3\end{array}\right.\) - step5: Substitute the value of \(y:\) \(-4-15-3z+z=3\) - step6: Simplify: \(-19-2z=3\) - step7: Move the constant to the right side: \(-2z=3+19\) - step8: Add the numbers: \(-2z=22\) - step9: Change the signs: \(2z=-22\) - step10: Divide both sides: \(\frac{2z}{2}=\frac{-22}{2}\) - step11: Divide the numbers: \(z=-11\) - step12: Substitute the value of \(z:\) \(y=-15-3\left(-11\right)\) - step13: Simplify: \(y=18\) - step14: Substitute the values of \(y,z:\) \(x=-4-18-\left(-11\right)\) - step15: Simplify: \(x=-11\) - step16: Calculate: \(\left\{ \begin{array}{l}x=-11\\y=18\\z=-11\end{array}\right.\) - step17: Check the solution: \(\left\{ \begin{array}{l}x=-11\\y=18\\z=-11\end{array}\right.\) - step18: Rewrite: \(\left(x,y,z\right) = \left(-11,18,-11\right)\) The solution to the system of equations using matrices (Gaussian elimination) is \( (x,y,z) = (-11,18,-11) \).