22. \( 3 x-y=2 \) and \( y=\frac{1}{3} x-11 \)

Solve the system of equations \( 3x-y=2;y=\frac{1}{3}x-11 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x-y=2\\y=\frac{1}{3}x-11\end{array}\right.\) - step1: Substitute the value of \(y:\) \(3x-\left(\frac{1}{3}x-11\right)=2\) - step2: Simplify: \(\frac{8}{3}x+11=2\) - step3: Move the constant to the right side: \(\frac{8}{3}x=2-11\) - step4: Subtract the numbers: \(\frac{8}{3}x=-9\) - step5: Multiply by the reciprocal: \(\frac{8}{3}x\times \frac{3}{8}=-9\times \frac{3}{8}\) - step6: Multiply: \(x=-\frac{27}{8}\) - step7: Substitute the value of \(x:\) \(y=\frac{1}{3}\left(-\frac{27}{8}\right)-11\) - step8: Calculate: \(y=-\frac{97}{8}\) - step9: Calculate: \(\left\{ \begin{array}{l}x=-\frac{27}{8}\\y=-\frac{97}{8}\end{array}\right.\) - step10: Check the solution: \(\left\{ \begin{array}{l}x=-\frac{27}{8}\\y=-\frac{97}{8}\end{array}\right.\) - step11: Rewrite: \(\left(x,y\right) = \left(-\frac{27}{8},-\frac{97}{8}\right)\) The solution to the system of equations \(3x-y=2\) and \(y=\frac{1}{3}x-11\) is \(x=-\frac{27}{8}\) and \(y=-\frac{97}{8}\) or \(x=-3.375\) and \(y=-12.125\).