1. Solve for \( x \) and \( y \) by using the method of substitution: (a) \( x-y=2 \) and \( 2 x+y=10 \) (b) \( y-3 x=-2 \) and \( 7 x-2 y=8+2 x- \) (c) \( 3 x+5 y=8 \) and \( x-2 y=-1 \) (d) \( 7 x-3 y=41 \) and \( 3 x-y=17 \quad 3 x= \) 2. Solve for \( x \) and \( y \) by using the method of elimination: (a) \( x-y=2 \) and \( 2 x+y=10 \) (b) \( x+y=-5 \) and \( 3 x+y=-9 \) (c) \( x+2 y=5 \) and \( x-y=-1 \) (d) \( 3 x+5 y=8 \) and \( x-2 y=-1 \) (c) \( 2 x-3 y=10 \) and \( 4 x+5 y=42 \)

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y=-5\\3x+y=-9\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-5-y\\3x+y=-9\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(-5-y\right)+y=-9\) - step3: Simplify: \(-15-2y=-9\) - step4: Move the constant to the right side: \(-2y=-9+15\) - step5: Add the numbers: \(-2y=6\) - step6: Change the signs: \(2y=-6\) - step7: Divide both sides: \(\frac{2y}{2}=\frac{-6}{2}\) - step8: Divide the numbers: \(y=-3\) - step9: Substitute the value of \(y:\) \(x=-5-\left(-3\right)\) - step10: Substitute back: \(x=-5+3\) - step11: Calculate: \(x=-2\) - step12: Calculate: \(\left\{ \begin{array}{l}x=-2\\y=-3\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}x=-2\\y=-3\end{array}\right.\) - step14: Rewrite: \(\left(x,y\right) = \left(-2,-3\right)\) Solve the system of equations \( 7 x-3 y=41;3 x-y=17 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}7x-3y=41\\3x-y=17\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}7x-3y=41\\y=-17+3x\end{array}\right.\) - step2: Substitute the value of \(y:\) \(7x-3\left(-17+3x\right)=41\) - step3: Simplify: \(-2x+51=41\) - step4: Move the constant to the right side: \(-2x=41-51\) - step5: Subtract the numbers: \(-2x=-10\) - step6: Change the signs: \(2x=10\) - step7: Divide both sides: \(\frac{2x}{2}=\frac{10}{2}\) - step8: Divide the numbers: \(x=5\) - step9: Substitute the value of \(x:\) \(y=-17+3\times 5\) - step10: Calculate: \(y=-2\) - step11: Calculate: \(\left\{ \begin{array}{l}x=5\\y=-2\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=5\\y=-2\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(5,-2\right)\) Solve the system of equations \( 2 x-3 y=10;4 x+5 y=42 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-3y=10\\4x+5y=42\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{10+3y}{2}\\4x+5y=42\end{array}\right.\) - step2: Substitute the value of \(x:\) \(4\times \frac{10+3y}{2}+5y=42\) - step3: Simplify: \(20+11y=42\) - step4: Move the constant to the right side: \(11y=42-20\) - step5: Subtract the numbers: \(11y=22\) - step6: Divide both sides: \(\frac{11y}{11}=\frac{22}{11}\) - step7: Divide the numbers: \(y=2\) - step8: Substitute the value of \(y:\) \(x=\frac{10+3\times 2}{2}\) - step9: Calculate: \(x=8\) - step10: Calculate: \(\left\{ \begin{array}{l}x=8\\y=2\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=8\\y=2\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(8,2\right)\) Solve the system of equations \( y-3 x=-2;7 x-2 y=8+2 x- \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y-3x=-2\\7x-2y=8+2x\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}y=-2+3x\\7x-2y=8+2x\end{array}\right.\) - step2: Substitute the value of \(y:\) \(7x-2\left(-2+3x\right)=8+2x\) - step3: Simplify: \(x+4=8+2x\) - step4: Move the expression to the left side: \(x-2x=8-4\) - step5: Add and subtract: \(-x=8-4\) - step6: Add and subtract: \(-x=4\) - step7: Change the signs: \(x=-4\) - step8: Substitute the value of \(x:\) \(y=-2+3\left(-4\right)\) - step9: Calculate: \(y=-14\) - step10: Calculate: \(\left\{ \begin{array}{l}x=-4\\y=-14\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=-4\\y=-14\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(-4,-14\right)\) Solve the system of equations \( x-y=2;2 x+y=10 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x-y=2\\2x+y=10\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=2+y\\2x+y=10\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(2+y\right)+y=10\) - step3: Simplify: \(4+3y=10\) - step4: Move the constant to the right side: \(3y=10-4\) - step5: Subtract the numbers: \(3y=6\) - step6: Divide both sides: \(\frac{3y}{3}=\frac{6}{3}\) - step7: Divide the numbers: \(y=2\) - step8: Substitute the value of \(y:\) \(x=2+2\) - step9: Calculate: \(x=4\) - step10: Calculate: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(4,2\right)\) Solve the system of equations \( x+2 y=5;x-y=-1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+2y=5\\x-y=-1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=5-2y\\x-y=-1\end{array}\right.\) - step2: Substitute the value of \(x:\) \(5-2y-y=-1\) - step3: Subtract the terms: \(5-3y=-1\) - step4: Move the constant to the right side: \(-3y=-1-5\) - step5: Subtract the numbers: \(-3y=-6\) - step6: Change the signs: \(3y=6\) - step7: Divide both sides: \(\frac{3y}{3}=\frac{6}{3}\) - step8: Divide the numbers: \(y=2\) - step9: Substitute the value of \(y:\) \(x=5-2\times 2\) - step10: Calculate: \(x=1\) - step11: Calculate: \(\left\{ \begin{array}{l}x=1\\y=2\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=1\\y=2\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(1,2\right)\) Solve the system of equations \( x-y=2;2 x+y=10 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x-y=2\\2x+y=10\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=2+y\\2x+y=10\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(2+y\right)+y=10\) - step3: Simplify: \(4+3y=10\) - step4: Move the constant to the right side: \(3y=10-4\) - step5: Subtract the numbers: \(3y=6\) - step6: Divide both sides: \(\frac{3y}{3}=\frac{6}{3}\) - step7: Divide the numbers: \(y=2\) - step8: Substitute the value of \(y:\) \(x=2+2\) - step9: Calculate: \(x=4\) - step10: Calculate: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(4,2\right)\) Solve the system of equations \( 3 x+5 y=8;x-2 y=-1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x+5y=8\\x-2y=-1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}3x+5y=8\\x=-1+2y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(-1+2y\right)+5y=8\) - step3: Simplify: \(-3+11y=8\) - step4: Move the constant to the right side: \(11y=8+3\) - step5: Add the numbers: \(11y=11\) - step6: Divide both sides: \(\frac{11y}{11}=\frac{11}{11}\) - step7: Divide the numbers: \(y=1\) - step8: Substitute the value of \(y:\) \(x=-1+2\times 1\) - step9: Substitute back: \(x=-1+2\) - step10: Calculate: \(x=1\) - step11: Calculate: \(\left\{ \begin{array}{l}x=1\\y=1\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=1\\y=1\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(1,1\right)\) Solve the system of equations \( 3 x+5 y=8;x-2 y=-1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x+5y=8\\x-2y=-1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}3x+5y=8\\x=-1+2y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(-1+2y\right)+5y=8\) - step3: Simplify: \(-3+11y=8\) - step4: Move the constant to the right side: \(11y=8+3\) - step5: Add the numbers: \(11y=11\) - step6: Divide both sides: \(\frac{11y}{11}=\frac{11}{11}\) - step7: Divide the numbers: \(y=1\) - step8: Substitute the value of \(y:\) \(x=-1+2\times 1\) - step9: Substitute back: \(x=-1+2\) - step10: Calculate: \(x=1\) - step11: Calculate: \(\left\{ \begin{array}{l}x=1\\y=1\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=1\\y=1\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(1,1\right)\) Here are the solutions for the systems of equations using both the substitution and elimination methods: ### 1. Solve for \( x \) and \( y \) by using the method of substitution: (a) \( x - y = 2 \) and \( 2x + y = 10 \) **Solution:** \( (x, y) = (4, 2) \) (b) \( y - 3x = -2 \) and \( 7x - 2y = 8 + 2x \) **Solution:** \( (x, y) = (-4, -14) \) (c) \( 3x + 5y = 8 \) and \( x - 2y = -1 \) **Solution:** \( (x, y) = (1, 1) \) (d) \( 7x - 3y = 41 \) and \( 3x - y = 17 \) **Solution:** \( (x, y) = (5, -2) \) ### 2. Solve for \( x \) and \( y \) by using the method of elimination: (a) \( x - y = 2 \) and \( 2x + y = 10 \) **Solution:** \( (x, y) = (4, 2) \) (b) \( x + y = -5 \) and \( 3x + y = -9 \) **Solution:** \( (x, y) = (-2, -3) \) (c) \( x + 2y = 5 \) and \( x - y = -1 \) **Solution:** \( (x, y) = (1, 2) \) (d) \( 3x + 5y = 8 \) and \( x - 2y = -1 \) **Solution:** \( (x, y) = (1, 1) \) (e) \( 2x - 3y = 10 \) and \( 4x + 5y = 42 \) **Solution:** \( (x, y) = (8, 2) \) If you need further explanations or details on any specific problem, feel free to ask!