1. **State the problem:** We are given angles at point R with expressions $(12x - 4)^\circ$, $(4y)^\circ$, $(5y)^\circ$, and a right angle of $90^\circ$. We need to find the values of $x$ and $y$. 2. **Use the angle sum rule:** The angles around point R form a full circle, so their sum is $360^\circ$. 3. **Write the equation:** $$ (12x - 4) + 4y + 5y + 90 = 360 $$ 4. **Simplify the equation:** $$ 12x - 4 + 9y + 90 = 360 $$ $$ 12x + 9y + 86 = 360 $$ 5. **Isolate terms:** $$ 12x + 9y = 360 - 86 $$ $$ 12x + 9y = 274 $$ 6. **Simplify by dividing by common factor 1 (no change):** $$ \cancel{12}x + \cancel{9}y = 274 $$ 7. **Use the right angle to find a relation between $x$ and $y$:** Since the right angle is $90^\circ$, and the angles $(4y)^\circ$ and $(5y)^\circ$ are adjacent to it, their sum plus $90$ must be $180^\circ$ (straight line): $$ 4y + 5y + 90 = 180 $$ $$ 9y + 90 = 180 $$ $$ 9y = 90 $$ $$ y = 10 $$ 8. **Substitute $y=10$ into the first equation:** $$ 12x + 9(10) = 274 $$ $$ 12x + 90 = 274 $$ $$ 12x = 274 - 90 $$ $$ 12x = 184 $$ $$ x = \frac{184}{12} $$ 9. **Simplify fraction:** $$ x = \frac{\cancel{184}^{\times 1}}{\cancel{12}^{\times 1}} = \frac{184}{12} = \frac{46}{3} \approx 15.33 $$ **Final answers:** $$ x = \frac{46}{3} $$ $$ y = 10 $$