1. **State the problem:** We want to find the number of days after July 1 when the average number of dogs and cats at the shelter is the same. 2. **Set up the equation:** The average number of dogs is given by $$y = -3x + 42$$ and the average number of cats is given by $$y = 4x + 7$$. 3. **Find when the averages are equal:** Set the two expressions equal to each other: $$-3x + 42 = 4x + 7$$ 4. **Solve for $x$:** Add $3x$ to both sides: $$\cancel{-3x} + 42 + 3x = 4x + 7 + 3x$$ which simplifies to $$42 = 7x + 7$$ Subtract 7 from both sides: $$42 - 7 = 7x + \cancel{7} - 7$$ which simplifies to $$35 = 7x$$ Divide both sides by 7: $$\frac{35}{\cancel{7}} = \frac{7x}{\cancel{7}}$$ which simplifies to $$5 = x$$ 5. **Interpret the result:** After 5 days, the average number of cats and dogs at the shelter is the same. **Final answer:** 5 days (Option C)