1. **State the problem:** We are given a diagram with angles expressed in terms of $x$ and $y$ in degrees. We need to find the values of $x$ and $y$. 2. **Identify the angles and relationships:** The triangle has angles $118^\circ$, $3x + y$, and $2x + y$. The sum of angles in a triangle is $180^\circ$, so: $$118 + (3x + y) + (2x + y) = 180$$ 3. **Simplify the triangle angle sum equation:** $$118 + 3x + y + 2x + y = 180$$ $$118 + 5x + 2y = 180$$ 4. **Isolate terms:** $$5x + 2y = 180 - 118$$ $$5x + 2y = 62 \quad \text{(Equation 1)}$$ 5. **Use the adjacent angles on the straight line:** The angles $3x + 4y$ and $5y$ are adjacent on a straight line, so their sum is $180^\circ$: $$ (3x + 4y) + 5y = 180 $$ 6. **Simplify this equation:** $$3x + 4y + 5y = 180$$ $$3x + 9y = 180 \quad \text{(Equation 2)}$$ 7. **Solve the system of equations:** Equation 1: $5x + 2y = 62$ Equation 2: $3x + 9y = 180$ Multiply Equation 1 by 9 and Equation 2 by 2 to eliminate $y$: $$9(5x + 2y) = 9(62) \Rightarrow 45x + 18y = 558$$ $$2(3x + 9y) = 2(180) \Rightarrow 6x + 18y = 360$$ 8. **Subtract the second from the first:** $$ (45x + 18y) - (6x + 18y) = 558 - 360 $$ $$ 45x - 6x + \cancel{18y} - \cancel{18y} = 198 $$ $$ 39x = 198 $$ 9. **Solve for $x$:** $$ x = \frac{198}{39} = \frac{198 \div 3}{39 \div 3} = \frac{66}{13} \approx 5.08 $$ 10. **Substitute $x$ back into Equation 1 to find $y$:** $$5x + 2y = 62$$ $$5 \times \frac{66}{13} + 2y = 62$$ $$\frac{330}{13} + 2y = 62$$ Convert 62 to fraction with denominator 13: $$\frac{330}{13} + 2y = \frac{806}{13}$$ Subtract $\frac{330}{13}$ from both sides: $$2y = \frac{806}{13} - \frac{330}{13} = \frac{476}{13}$$ Divide both sides by 2: $$y = \frac{476}{13} \times \frac{1}{2} = \frac{476}{26} = \frac{238}{13} \approx 18.31$$ **Final answers:** $$x = \frac{66}{13} \approx 5.08, \quad y = \frac{238}{13} \approx 18.31$$