\( (x-2 y)(2 x+y)=12 \) and \( 7 y-x+9=0 \)

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\left(x-2y\right)\left(2x+y\right)-12=0\\7y-x+9=0\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}\left(x-2y\right)\left(2x+y\right)-12=0\\x=7y+9\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left(7y+9-2y\right)\left(2\left(7y+9\right)+y\right)-12=0\) - step3: Simplify: \(75y^{2}+225y+150=0\) - step4: Factor the expression: \(75\left(y+1\right)\left(y+2\right)=0\) - step5: Divide the terms: \(\left(y+1\right)\left(y+2\right)=0\) - step6: Separate into possible cases: \(\begin{align}&y+1=0\\&y+2=0\end{align}\) - step7: Solve the equation: \(\begin{align}&y=-1\\&y=-2\end{align}\) - step8: Calculate: \(y=-1\cup y=-2\) - step9: Rearrange the terms: \(\left\{ \begin{array}{l}x=7y+9\\y=-1\end{array}\right.\cup \left\{ \begin{array}{l}x=7y+9\\y=-2\end{array}\right.\) - step10: Calculate: \(\left\{ \begin{array}{l}x=2\\y=-1\end{array}\right.\cup \left\{ \begin{array}{l}x=-5\\y=-2\end{array}\right.\) - step11: Calculate: \(\left\{ \begin{array}{l}x=-5\\y=-2\end{array}\right.\cup \left\{ \begin{array}{l}x=2\\y=-1\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=-5\\y=-2\end{array}\right.\cup \left\{ \begin{array}{l}x=2\\y=-1\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(-5,-2\right)\cup \left(x,y\right) = \left(2,-1\right)\) To solve the system of equations given by: 1. \( (x-2y)(2x+y) = 12 \) 2. \( 7y - x + 9 = 0 \) we can find the solutions step by step. ### Step 1: Rewrite the equations The first equation can be rewritten as: \[ (x-2y)(2x+y) - 12 = 0 \] The second equation can be rearranged to express \( x \) in terms of \( y \): \[ x = 7y + 9 \] ### Step 2: Substitute \( x \) in the first equation Now, substitute \( x = 7y + 9 \) into the first equation: \[ ((7y + 9) - 2y)(2(7y + 9) + y) = 12 \] This simplifies to: \[ (5y + 9)(14y + 18 + y) = 12 \] \[ (5y + 9)(15y + 18) = 12 \] ### Step 3: Expand and simplify Now, expand the left side: \[ 75y^2 + 90y + 135y + 162 = 12 \] Combine like terms: \[ 75y^2 + 225y + 162 - 12 = 0 \] \[ 75y^2 + 225y + 150 = 0 \] ### Step 4: Factor or use the quadratic formula We can simplify this equation by dividing everything by 75: \[ y^2 + 3y + 2 = 0 \] Now, factor the quadratic: \[ (y + 1)(y + 2) = 0 \] ### Step 5: Solve for \( y \) Setting each factor to zero gives us: \[ y + 1 = 0 \quad \Rightarrow \quad y = -1 \] \[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \] ### Step 6: Find corresponding \( x \) values Now, substitute these \( y \) values back into \( x = 7y + 9 \): 1. For \( y = -1 \): \[ x = 7(-1) + 9 = 2 \] 2. For \( y = -2 \): \[ x = 7(-2) + 9 = -5 \] ### Final Solutions The solutions to the system of equations are: \[ (x, y) = (2, -1) \quad \text{and} \quad (x, y) = (-5, -2) \]