Question: 8) \(\frac{x}{2} - \frac{4y}{3} = 5\) \(\frac{5x}{2} + y = 2\)

8. \(\frac{x}{2} - \frac{4y}{3} = 5\)

\(\frac{5x}{2} + y = 2\)

Solution

To solve the system of equations, we have: \[ \begin{cases} \frac{x}{2} - \frac{4y}{3} = 5 & \\ \frac{5x}{2} + y = 2 & \end{cases} \] Step 1: Solve the second equation for \( y \). \[ y = 2 - \frac{5x}{2} \] Step 2: Substitute \( y = 2 - \frac{5x}{2} \) into the first equation. \[ \frac{x}{2} - \frac{4}{3} \left(2 - \frac{5x}{2}\right) = 5 \] Step 3: Simplify the equation. First, distribute \(-\frac{4}{3}\) through the parenthesis: \[ \frac{x}{2} - \frac{8}{3} + \frac{20x}{6} = 5 \] Step 4: Convert \(\frac{20x}{6}\) to \(\frac{10x}{3}\) since they are equivalent. Step 5: Convert \(\frac{x}{2}\) to \(\frac{3x}{6}\) to have a common denominator with \(\frac{10x}{3}\). Step 6: Rewrite the equation with a common denominator: \[ \frac{3x}{6} + \frac{10x}{3} - \frac{8}{3} = 5 \] Convert \(\frac{3x}{6}\) to \(\frac{x}{2}\) and \(\frac{10x}{3}\) to \(\frac{20x}{6}\) for the common denominator: \[ \frac{3x}{6} + \frac{20x}{6} = 5 + \frac{8}{3} \] Combine \(\frac{3x}{6}\) and \(\frac{20x}{6}\): \[ \frac{23x}{6} = 5 + \frac{8}{3} \] Step 7: Convert 5 to \(\frac{15}{3}\): \[ \frac{23x}{6} = \frac{15}{3} + \frac{8}{3} \] \[ \frac{23x}{6} = \frac{23}{3} \] Step 8: Solve for \( x \). Multiply both sides by 6: \[ 23x = 46 \] Divide by 23: \[ x = 2 \] Step 9: Substitute \( x = 2 \) back into \( y = 2 - \frac{5x}{2} \). \[ y = 2 - \frac{5 \cdot 2}{2} \] \[ y = 2 - 5 \] \[ y = -3 \] The solution to the system of equations is \( x = 2 \) and \( y = -3 \).