1. **State the problem:** We have a triangle with angles labeled as follows: the top vertex angle is 80 degrees, the left base angle is $2x - 6$, and the right base angle is $3x + 1$. We need to find the value of $y$, which is the vertex angle opposite the base. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always 180 degrees. Therefore, we have: $$ (2x - 6) + (3x + 1) + 80 = 180 $$ 3. **Simplify the equation:** $$ 2x - 6 + 3x + 1 + 80 = 180 $$ $$ 5x + 75 = 180 $$ 4. **Solve for $x$:** $$ 5x = 180 - 75 $$ $$ 5x = 105 $$ $$ x = \frac{105}{5} = 21 $$ 5. **Find the base angles:** Left base angle: $$ 2x - 6 = 2(21) - 6 = 42 - 6 = 36 $$ Right base angle: $$ 3x + 1 = 3(21) + 1 = 63 + 1 = 64 $$ 6. **Find $y$:** Since $y$ is the vertex angle opposite the base, and the triangle's angles sum to 180 degrees, we calculate: $$ y = 180 - (36 + 64) = 180 - 100 = 80 $$ **Final answer:** $$ y = 80 $$