1. **State the problem:** We are given a triangle with angles expressed as $3x - 5$, $5y - 4$, and $y + 12$. We need to find the values of $x$ and $y$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. So, $$ (3x - 5) + (5y - 4) + (y + 12) = 180 $$ 3. **Combine like terms:** $$ 3x - 5 + 5y - 4 + y + 12 = 180 $$ $$ 3x + 6y + ( -5 -4 + 12 ) = 180 $$ $$ 3x + 6y + 3 = 180 $$ 4. **Isolate terms:** $$ 3x + 6y = 180 - 3 $$ $$ 3x + 6y = 177 $$ 5. **Simplify by dividing both sides by 3:** $$ \cancel{3}x + \cancel{3} \times 2 y = \cancel{3} \times 59 $$ $$ x + 2y = 59 $$ 6. **Express $x$ in terms of $y$:** $$ x = 59 - 2y $$ 7. **Use the smaller triangle's angle relationships:** Since the smaller triangle shares an angle and the problem implies a relationship, assume the smaller triangle's angles sum to $180^\circ$ and relate to the larger triangle's angles. However, without additional equations or information, we cannot solve for both $x$ and $y$ uniquely. **Conclusion:** The problem as stated provides only one equation with two variables. To find unique values for $x$ and $y$, more information or equations are needed. **Final answer:** $$ x = 59 - 2y $$ Without additional data, $y$ remains a free variable.