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

To solve these two equations, you can first express one variable in terms of the other from the first equation and then substitute it into the second equation. From the first equation, \( \frac{3}{4} x = \frac{1}{2} y \) This gives \( y = \frac{3}{2} x \). Now you can substitute this expression for \( y \) into the second equation: \( \frac{1}{2} x + \frac{1}{3} \left(\frac{3}{2} x\right) = 4 \). This simplifies to \( \frac{1}{2} x + \frac{1}{2} x = 4 \), or \( x = 4 \). Now substitute \( x = 4 \) back into \( y = \frac{3}{2} x \) to find \( y \): \( y = \frac{3}{2} \times 4 = 6 \). So, the solution to the system is \( x = 4 \) and \( y = 6 \). If you want to check your work, you can substitute these values back into the original equations to verify that they hold true!