1. **State the problem:** We are given two adjacent angles on a straight line, labeled $(-7y + 97)^\circ$ and $(3y + 75)^\circ$, and we need to find the value of $y$. 2. **Use the supplementary angles rule:** Adjacent angles on a straight line add up to $180^\circ$. So, $$(-7y + 97) + (3y + 75) = 180$$ 3. **Combine like terms:** $$-7y + 3y + 97 + 75 = 180$$ $$-4y + 172 = 180$$ 4. **Isolate $y$:** $$-4y + 172 = 180$$ $$-4y = 180 - 172$$ $$-4y = 8$$ 5. **Divide both sides by $-4$:** $$y = \frac{8}{\cancel{-4}} \times \cancel{-1} = -2$$ **Final answer:** $y = -2$