\( \frac{3}{4} x-\frac{1}{2} y=0 \) and \( \frac{1}{2} x+\frac{1}{3} y=4 \)

Solve the system of equations \( \frac{3}{4}x-\frac{1}{2}y=0; \frac{1}{2}x+\frac{1}{3}y=4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\frac{3}{4}x-\frac{1}{2}y=0\\\frac{1}{2}x+\frac{1}{3}y=4\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=\frac{2y}{3}\\\frac{1}{2}x+\frac{1}{3}y=4\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\frac{1}{2}\times \frac{2y}{3}+\frac{1}{3}y=4\) - step3: Multiply the terms: \(\frac{y}{3}+\frac{1}{3}y=4\) - step4: Multiply both sides of the equation by LCD: \(\left(\frac{y}{3}+\frac{1}{3}y\right)\times 3=4\times 3\) - step5: Simplify the equation: \(2y=12\) - step6: Divide both sides: \(\frac{2y}{2}=\frac{12}{2}\) - step7: Divide the numbers: \(y=6\) - step8: Substitute the value of \(y:\) \(x=\frac{2\times 6}{3}\) - step9: Calculate: \(x=4\) - step10: Calculate: \(\left\{ \begin{array}{l}x=4\\y=6\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=6\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(4,6\right)\) The solution to the system of equations is \( x = 4 \) and \( y = 6 \).