1. Solve the pairs of simultaneous equations. a) \( x+y=5 ; \quad x-y=3 \) b) \( x-3 y=-5 ; \quad 2 x+3 y=-1 \) c) \( 2 x-3 y=-21 ; \quad 2 x+3 y=1 \) c)

Solve the equation \( x+y=5;x-y=3 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y=5\\x-y=3\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=5-y\\x-y=3\end{array}\right.\) - step2: Substitute the value of \(x:\) \(5-y-y=3\) - step3: Subtract the terms: \(5-2y=3\) - step4: Move the constant to the right side: \(-2y=3-5\) - step5: Subtract the numbers: \(-2y=-2\) - step6: Change the signs: \(2y=2\) - step7: Divide both sides: \(\frac{2y}{2}=\frac{2}{2}\) - step8: Divide the numbers: \(y=1\) - step9: Substitute the value of \(y:\) \(x=5-1\) - step10: Calculate: \(x=4\) - step11: Calculate: \(\left\{ \begin{array}{l}x=4\\y=1\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=1\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(4,1\right)\) Solve the equation \( x-3y=-5;2x+3y=-1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x-3y=-5\\2x+3y=-1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-5+3y\\2x+3y=-1\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(-5+3y\right)+3y=-1\) - step3: Simplify: \(-10+9y=-1\) - step4: Move the constant to the right side: \(9y=-1+10\) - step5: Add the numbers: \(9y=9\) - step6: Divide both sides: \(\frac{9y}{9}=\frac{9}{9}\) - step7: Divide the numbers: \(y=1\) - step8: Substitute the value of \(y:\) \(x=-5+3\times 1\) - step9: Substitute back: \(x=-5+3\) - step10: Calculate: \(x=-2\) - step11: Calculate: \(\left\{ \begin{array}{l}x=-2\\y=1\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=-2\\y=1\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(-2,1\right)\) Solve the equation \( 2x-3y=-21;2x+3y=1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-3y=-21\\2x+3y=1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{-21+3y}{2}\\2x+3y=1\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\times \frac{-21+3y}{2}+3y=1\) - step3: Simplify: \(-21+6y=1\) - step4: Move the constant to the right side: \(6y=1+21\) - step5: Add the numbers: \(6y=22\) - step6: Divide both sides: \(\frac{6y}{6}=\frac{22}{6}\) - step7: Divide the numbers: \(y=\frac{11}{3}\) - step8: Substitute the value of \(y:\) \(x=\frac{-21+3\times \frac{11}{3}}{2}\) - step9: Calculate: \(x=-5\) - step10: Calculate: \(\left\{ \begin{array}{l}x=-5\\y=\frac{11}{3}\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=-5\\y=\frac{11}{3}\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(-5,\frac{11}{3}\right)\) The solutions to the pairs of simultaneous equations are: a) \( x = 4, y = 1 \) b) \( x = -2, y = 1 \) c) \( x = -5, y = \frac{11}{3} \) or \( x = -5, y = 3.6 \)