Solve for \( x \) and \( y \) simultanously (1) \( 3 x+2 y=7 \) and \( -3 x+y=1 \) (2) \( x+3 y=17 \) and \( 4 x-2 y=12 \) (3) \( x+2 y=13 \) and \( 3 x-y=4 \)

Solve the system of equations \( 3x+2y=7;-3x+y=1 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3x+2y=7\\-3x+y=1\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}3x+2y=7\\y=1+3x\end{array}\right.\) - step2: Substitute the value of \(y:\) \(3x+2\left(1+3x\right)=7\) - step3: Simplify: \(9x+2=7\) - step4: Move the constant to the right side: \(9x=7-2\) - step5: Subtract the numbers: \(9x=5\) - step6: Divide both sides: \(\frac{9x}{9}=\frac{5}{9}\) - step7: Divide the numbers: \(x=\frac{5}{9}\) - step8: Substitute the value of \(x:\) \(y=1+3\times \frac{5}{9}\) - step9: Calculate: \(y=\frac{8}{3}\) - step10: Calculate: \(\left\{ \begin{array}{l}x=\frac{5}{9}\\y=\frac{8}{3}\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=\frac{5}{9}\\y=\frac{8}{3}\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(\frac{5}{9},\frac{8}{3}\right)\) Solve the system of equations \( x+3y=17;4x-2y=12 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+3y=17\\4x-2y=12\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=17-3y\\4x-2y=12\end{array}\right.\) - step2: Substitute the value of \(x:\) \(4\left(17-3y\right)-2y=12\) - step3: Simplify: \(68-14y=12\) - step4: Move the constant to the right side: \(-14y=12-68\) - step5: Subtract the numbers: \(-14y=-56\) - step6: Change the signs: \(14y=56\) - step7: Divide both sides: \(\frac{14y}{14}=\frac{56}{14}\) - step8: Divide the numbers: \(y=4\) - step9: Substitute the value of \(y:\) \(x=17-3\times 4\) - step10: Calculate: \(x=5\) - step11: Calculate: \(\left\{ \begin{array}{l}x=5\\y=4\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=5\\y=4\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(5,4\right)\) Solve the system of equations \( x+2y=13;3x-y=4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+2y=13\\3x-y=4\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=13-2y\\3x-y=4\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(13-2y\right)-y=4\) - step3: Simplify: \(39-7y=4\) - step4: Move the constant to the right side: \(-7y=4-39\) - step5: Subtract the numbers: \(-7y=-35\) - step6: Change the signs: \(7y=35\) - step7: Divide both sides: \(\frac{7y}{7}=\frac{35}{7}\) - step8: Divide the numbers: \(y=5\) - step9: Substitute the value of \(y:\) \(x=13-2\times 5\) - step10: Calculate: \(x=3\) - step11: Calculate: \(\left\{ \begin{array}{l}x=3\\y=5\end{array}\right.\) - step12: Check the solution: \(\left\{ \begin{array}{l}x=3\\y=5\end{array}\right.\) - step13: Rewrite: \(\left(x,y\right) = \left(3,5\right)\) The solutions to the given systems of equations are: 1. \( x = \frac{5}{9} \) and \( y = \frac{8}{3} \) 2. \( x = 5 \) and \( y = 4 \) 3. \( x = 3 \) and \( y = 5 \)