52. \( \left\{\begin{aligned} 2 x+3 y & =4 \\ -x+2 y & =-2\end{aligned}\right. \)

Solve the system of equations \( 2x+3y=4;-x+2y=-2 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x+3y=4\\-x+2y=-2\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}2x+3y=4\\x=2+2y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(2+2y\right)+3y=4\) - step3: Simplify: \(4+7y=4\) - step4: Move the constant to the right side: \(7y=4-4\) - step5: Subtract the terms: \(7y=0\) - step6: Rewrite the expression: \(y=0\) - step7: Substitute the value of \(y:\) \(x=2+2\times 0\) - step8: Substitute back: \(x=2\) - step9: Calculate: \(\left\{ \begin{array}{l}x=2\\y=0\end{array}\right.\) - step10: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=0\end{array}\right.\) - step11: Rewrite: \(\left(x,y\right) = \left(2,0\right)\) The solution to the system of equations is \( x = 2 \) and \( y = 0 \).