Solve simcultanecusily \( x+2 y=5 \) and \( 2 y^{2}-x y-4 x^{2}=8 \)

Solve the system of equations \( x+2y=5;2y^{2}-xy-4x^{2}=8 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+2y=5\\2y^{2}-xy-4x^{2}=8\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=5-2y\\2y^{2}-xy-4x^{2}=8\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2y^{2}-\left(5-2y\right)y-4\left(5-2y\right)^{2}=8\) - step3: Simplify: \(-12y^{2}+75y-100=8\) - step4: Move the expression to the left side: \(-12y^{2}+75y-100-8=0\) - step5: Subtract the numbers: \(-12y^{2}+75y-108=0\) - step6: Factor the expression: \(-3\left(y-4\right)\left(4y-9\right)=0\) - step7: Divide the terms: \(\left(y-4\right)\left(4y-9\right)=0\) - step8: Separate into possible cases: \(\begin{align}&y-4=0\\&4y-9=0\end{align}\) - step9: Solve the equation: \(\begin{align}&y=4\\&y=\frac{9}{4}\end{align}\) - step10: Calculate: \(y=4\cup y=\frac{9}{4}\) - step11: Rearrange the terms: \(\left\{ \begin{array}{l}x=5-2y\\y=4\end{array}\right.\cup \left\{ \begin{array}{l}x=5-2y\\y=\frac{9}{4}\end{array}\right.\) - step12: Calculate: \(\left\{ \begin{array}{l}x=-3\\y=4\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{1}{2}\\y=\frac{9}{4}\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}x=-3\\y=4\end{array}\right.\cup \left\{ \begin{array}{l}x=\frac{1}{2}\\y=\frac{9}{4}\end{array}\right.\) - step14: Rewrite: \(\left(x,y\right) = \left(-3,4\right)\cup \left(x,y\right) = \left(\frac{1}{2},\frac{9}{4}\right)\) The solutions to the system of equations \(x+2y=5\) and \(2y^{2}-xy-4x^{2}=8\) are: 1. \(x = -3, y = 4\) 2. \(x = 0.5, y = 2.25\)