\( x+4 y=9 \) and \( y=-\frac{1}{x-1}+2 \)

Solve the system of equations \( x+4y=9;y=-\frac{1}{x-1}+2 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+4y=9\\y=-\frac{1}{x-1}+2\end{array}\right.\) - step1: Substitute the value of \(y:\) \(x+4\left(-\frac{1}{x-1}+2\right)=9\) - step2: Multiply the terms: \(x+\frac{-12+8x}{x-1}=9\) - step3: Multiply both sides of the equation by LCD: \(\left(x+\frac{-12+8x}{x-1}\right)\left(x-1\right)=9\left(x-1\right)\) - step4: Simplify the equation: \(x^{2}+7x-12=9x-9\) - step5: Move the expression to the left side: \(x^{2}+7x-12-\left(9x-9\right)=0\) - step6: Calculate: \(x^{2}-2x-3=0\) - step7: Factor the expression: \(\left(x-3\right)\left(x+1\right)=0\) - step8: Separate into possible cases: \(\begin{align}&x-3=0\\&x+1=0\end{align}\) - step9: Solve the equation: \(\begin{align}&x=3\\&x=-1\end{align}\) - step10: Calculate: \(x=3\cup x=-1\) - step11: Rearrange the terms: \(\left\{ \begin{array}{l}x=3\\y=-\frac{1}{x-1}+2\end{array}\right.\cup \left\{ \begin{array}{l}x=-1\\y=-\frac{1}{x-1}+2\end{array}\right.\) - step12: Calculate: \(\left\{ \begin{array}{l}x=3\\y=\frac{3}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=-1\\y=\frac{5}{2}\end{array}\right.\) - step13: Calculate: \(\left\{ \begin{array}{l}x=-1\\y=\frac{5}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=3\\y=\frac{3}{2}\end{array}\right.\) - step14: Check the solution: \(\left\{ \begin{array}{l}x=-1\\y=\frac{5}{2}\end{array}\right.\cup \left\{ \begin{array}{l}x=3\\y=\frac{3}{2}\end{array}\right.\) - step15: Rewrite: \(\left(x,y\right) = \left(-1,\frac{5}{2}\right)\cup \left(x,y\right) = \left(3,\frac{3}{2}\right)\) The system of equations \( x+4y=9 \) and \( y=-\frac{1}{x-1}+2 \) has two solutions: 1. \( x = -1 \) and \( y = \frac{5}{2} \) 2. \( x = 3 \) and \( y = \frac{3}{2} \)