Sowe for \( x \) and \( y \) simutanously \( 2 x-y=8 \) \( y=x^{2}+4 x-23 \) 12

Solve the system of equations \( 2x-y=8;y=x^{2}+4x-23 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-y=8\\y=x^{2}+4x-23\end{array}\right.\) - step1: Substitute the value of \(y:\) \(2x-\left(x^{2}+4x-23\right)=8\) - step2: Simplify: \(-2x-x^{2}+23=8\) - step3: Move the expression to the left side: \(-2x-x^{2}+23-8=0\) - step4: Subtract the numbers: \(-2x-x^{2}+15=0\) - step5: Factor the expression: \(\left(-x+3\right)\left(x+5\right)=0\) - step6: Separate into possible cases: \(\begin{align}&-x+3=0\\&x+5=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=3\\&x=-5\end{align}\) - step8: Calculate: \(x=3\cup x=-5\) - step9: Rearrange the terms: \(\left\{ \begin{array}{l}x=3\\y=x^{2}+4x-23\end{array}\right.\cup \left\{ \begin{array}{l}x=-5\\y=x^{2}+4x-23\end{array}\right.\) - step10: Calculate: \(\left\{ \begin{array}{l}x=3\\y=-2\end{array}\right.\cup \left\{ \begin{array}{l}x=-5\\y=-18\end{array}\right.\) - step11: Calculate: \(\left\{ \begin{array}{l}x=-5\\y=-18\end{array}\right.\cup \left\{ \begin{array}{l}x=3\\y=-2\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=-5\\y=-18\end{array}\right.\cup \left\{ \begin{array}{l}x=3\\y=-2\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(-5,-18\right)\cup \left(x,y\right) = \left(3,-2\right)\) The solutions to the system of equations \(2x-y=8\) and \(y=x^{2}+4x-23\) are \((-5,-18)\) and \((3,-2)\).