c) \( \left\{\begin{array}{l}\frac{1}{2} x+\frac{3}{2} y=5 \\ 3 x-y=5 y\end{array}\right. \)

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\frac{1}{2}x+\frac{3}{2}y=5\\3x-y=5y\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=10-3y\\3x-y=5y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(3\left(10-3y\right)-y=5y\) - step3: Simplify: \(30-10y=5y\) - step4: Move the variable to the left side: \(30-10y-5y=0\) - step5: Subtract the terms: \(30-15y=0\) - step6: Move the constant to the right side: \(-15y=0-30\) - step7: Remove 0: \(-15y=-30\) - step8: Change the signs: \(15y=30\) - step9: Divide both sides: \(\frac{15y}{15}=\frac{30}{15}\) - step10: Divide the numbers: \(y=2\) - step11: Substitute the value of \(y:\) \(x=10-3\times 2\) - step12: Calculate: \(x=4\) - step13: Calculate: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step14: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step15: Rewrite: \(\left(x,y\right) = \left(4,2\right)\) La solución del sistema de ecuaciones es \( (x, y) = (4, 2) \). Esto significa que \( x = 4 \) y \( y = 2 \).