1. **State the problem:** We are given the equation of a line $y = -8x + 8$ and a table showing possible combinations of chickens ($x$) and cows ($y$) on a farm, where the total number of legs is 82. We want to understand how these combinations relate to the equation and verify the values. 2. **Understand the context:** Each chicken has 2 legs and each cow has 4 legs. The total legs equation is: $$2x + 4y = 82$$ where $x$ is the number of chickens and $y$ is the number of cows. 3. **Rewrite the total legs equation to express $y$ in terms of $x$:** $$2x + 4y = 82$$ Subtract $2x$ from both sides: $$4y = 82 - 2x$$ Divide both sides by 4: $$y = \frac{82 - 2x}{4}$$ Show cancellation: $$y = \frac{\cancel{2} \times 41 - \cancel{2} x}{\cancel{4} \times 2} = \frac{41 - x}{2}$$ 4. **Compare this with the given line $y = -8x + 8$:** The given line does not represent the leg count equation. The leg count line is: $$y = \frac{41 - x}{2} = 20.5 - 0.5x$$ 5. **Check the table values against the leg count equation:** - For $x=35$, $y = \frac{41 - 35}{2} = \frac{6}{2} = 3$ but table shows 7 cows, so this does not satisfy the leg count equation. - For $x=10$, $y = \frac{41 - 10}{2} = \frac{31}{2} = 15.5$ but table shows 5 cows, so no. - For $x=19$, $y = \frac{41 - 19}{2} = \frac{22}{2} = 11$ but table shows 2 cows, so no. 6. **Conclusion:** The table values do not satisfy the leg count equation $2x + 4y = 82$. The given line $y = -8x + 8$ is unrelated to the leg count problem. **Final answer:** The correct equation relating chickens and cows for 82 legs is: $$y = \frac{41 - x}{2}$$ The given line $y = -8x + 8$ does not represent this relationship.