EXERCISE 9 Determine the value of \( x \) and \( y \) which satisfy the following systems of linear cquations: \( \begin{array}{lll}\text { (a) } x-y=2 \text { and } 2 x+y=10 & \text { (b) } y-3 x=-2 \text { and } 7 x-2 y=8 \\ \text { (c) } 3 x+5 y=8 \text { and } x-2 y=-1 & \text { (d) } 7 x-3 y=41 \text { and } 3 x-y=17\end{array} \)

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 \( y-3 x=-2;7 x-2 y=8 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y-3x=-2\\7x-2y=8\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}y=-2+3x\\7x-2y=8\end{array}\right.\) - step2: Substitute the value of \(y:\) \(7x-2\left(-2+3x\right)=8\) - step3: Simplify: \(x+4=8\) - step4: Move the constant to the right side: \(x=8-4\) - step5: Subtract the numbers: \(x=4\) - step6: Substitute the value of \(x:\) \(y=-2+3\times 4\) - step7: Calculate: \(y=10\) - step8: Calculate: \(\left\{ \begin{array}{l}x=4\\y=10\end{array}\right.\) - step9: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=10\end{array}\right.\) - step10: Rewrite: \(\left(x,y\right) = \left(4,10\right)\) 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 \( x-y=2;2 x+y=10 \). 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: \(\left(x,y\right) = \left(4,2\right)\) Here are the solutions for each system of linear equations: ### (a) Equations: 1. \( x - y = 2 \) 2. \( 2x + y = 10 \) **Solution:** \[ (x, y) = (4, 2) \] ### (b) Equations: 1. \( y - 3x = -2 \) 2. \( 7x - 2y = 8 \) **Solution:** \[ (x, y) = (4, 10) \] ### (c) Equations: 1. \( 3x + 5y = 8 \) 2. \( x - 2y = -1 \) **Solution:** \[ (x, y) = (1, 1) \] ### (d) Equations: 1. \( 7x - 3y = 41 \) 2. \( 3x - y = 17 \) **Solution:** \[ (x, y) = (5, -2) \] These values of \( x \) and \( y \) satisfy their respective systems of equations.