1. \( 4 x-5 y=-20 \) and \( 2 x+y=4 \) 2. \( 2 x-3 y=2 \) and \( 5 x+y=5 \) 3. \( x+3 y=12 \) and \( 2 x+y=9 \) 4. \( 3 x-5 y=4 \) and \( 2 x-3 y=7 \) 5. \( 2 x-3 y=12 \) and \( x+2 y=6 \)

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-3y=12\\x+2y=6\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}2x-3y=12\\x=6-2y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(6-2y\right)-3y=12\) - step3: Simplify: \(12-7y=12\) - step4: Move the constant to the right side: \(-7y=12-12\) - step5: Subtract the terms: \(-7y=0\) - step6: Change the signs: \(7y=0\) - step7: Rewrite the expression: \(y=0\) - step8: Substitute the value of \(y:\) \(x=6-2\times 0\) - step9: Substitute back: \(x=6\) - step10: Calculate: \(\left\{ \begin{array}{l}x=6\\y=0\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=6\\y=0\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(6,0\right)\) Solve the system of equations \( 4 x-5 y=-20; 2 x+y=4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}4x-5y=-20\\2x+y=4\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}4x-5y=-20\\y=4-2x\end{array}\right.\) - step2: Substitute the value of \(y:\) \(4x-5\left(4-2x\right)=-20\) - step3: Simplify: \(14x-20=-20\) - step4: Move the constant to the right side: \(14x=-20+20\) - step5: Add the numbers: \(14x=0\) - step6: Rewrite the expression: \(x=0\) - step7: Substitute the value of \(x:\) \(y=4-2\times 0\) - step8: Substitute back: \(y=4\) - step9: Calculate: \(\left\{ \begin{array}{l}x=0\\y=4\end{array}\right.\) - step10: Check the solution: \(\left\{ \begin{array}{l}x=0\\y=4\end{array}\right.\) - step11: Rewrite: \(\left(x,y\right) = \left(0,4\right)\) Solve the system of equations \( 2 x-3 y=2; 5 x+y=5 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-3y=2\\5x+y=5\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}2x-3y=2\\y=5-5x\end{array}\right.\) - step2: Substitute the value of \(y:\) \(2x-3\left(5-5x\right)=2\) - step3: Simplify: \(17x-15=2\) - step4: Move the constant to the right side: \(17x=2+15\) - step5: Add the numbers: \(17x=17\) - step6: Divide both sides: \(\frac{17x}{17}=\frac{17}{17}\) - step7: Divide the numbers: \(x=1\) - step8: Substitute the value of \(x:\) \(y=5-5\times 1\) - step9: Substitute back: \(y=5-5\) - step10: Calculate: \(y=0\) - step11: Calculate: \(\left\{ \begin{array}{l}x=1\\y=0\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=1\\y=0\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(1,0\right)\) Solve the system of equations \( x+3 y=12; 2 x+y=9 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+3y=12\\2x+y=9\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=12-3y\\2x+y=9\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(12-3y\right)+y=9\) - step3: Simplify: \(24-5y=9\) - step4: Move the constant to the right side: \(-5y=9-24\) - step5: Subtract the numbers: \(-5y=-15\) - step6: Change the signs: \(5y=15\) - step7: Divide both sides: \(\frac{5y}{5}=\frac{15}{5}\) - step8: Divide the numbers: \(y=3\) - step9: Substitute the value of \(y:\) \(x=12-3\times 3\) - step10: Calculate: \(x=3\) - step11: Calculate: \(\left\{ \begin{array}{l}x=3\\y=3\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=3\\y=3\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(3,3\right)\) Solve the system of equations \( 3 x-5 y=4; 2 x-3 y=7 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x-5y=4\\2x-3y=7\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{4+5y}{3}\\2x-3y=7\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\times \frac{4+5y}{3}-3y=7\) - step3: Simplify: \(\frac{2\left(4+5y\right)}{3}-3y=7\) - step4: Multiply both sides of the equation by LCD: \(\left(\frac{2\left(4+5y\right)}{3}-3y\right)\times 3=7\times 3\) - step5: Simplify the equation: \(8+y=21\) - step6: Move the constant to the right side: \(y=21-8\) - step7: Subtract the numbers: \(y=13\) - step8: Substitute the value of \(y:\) \(x=\frac{4+5\times 13}{3}\) - step9: Calculate: \(x=23\) - step10: Calculate: \(\left\{ \begin{array}{l}x=23\\y=13\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=23\\y=13\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(23,13\right)\) Here are the solutions for the given systems of equations: 1. For the equations \( 4x - 5y = -20 \) and \( 2x + y = 4 \): - The solution is \( (x, y) = (0, 4) \). 2. For the equations \( 2x - 3y = 2 \) and \( 5x + y = 5 \): - The solution is \( (x, y) = (1, 0) \). 3. For the equations \( x + 3y = 12 \) and \( 2x + y = 9 \): - The solution is \( (x, y) = (3, 3) \). 4. For the equations \( 3x - 5y = 4 \) and \( 2x - 3y = 7 \): - The solution is \( (x, y) = (23, 13) \). 5. For the equations \( 2x - 3y = 12 \) and \( x + 2y = 6 \): - The solution is \( (x, y) = (6, 0) \). If you need further assistance or explanations for any of these solutions, feel free to ask!