1. **State the problem:** We have chickens and cows on a farm. Chickens have 2 legs each, cows have 4 legs each, and the total number of legs is 82. 2. **Define variables:** Let $x$ be the number of chickens and $y$ be the number of cows. 3. **Write the equation:** Since chickens have 2 legs and cows have 4 legs, the total legs equation is: $$2x + 4y = 82$$ 4. **Simplify the equation:** Divide both sides by 2: $$\cancel{2}x + \cancel{4}y = \cancel{82}$$ $$x + 2y = 41$$ 5. **Use the equation to find missing values in the table:** - For $x=35$, find $y$: $$35 + 2y = 41 \Rightarrow 2y = 41 - 35 = 6 \Rightarrow y = 3$$ - For $y=7$, find $x$: $$x + 2(7) = 41 \Rightarrow x + 14 = 41 \Rightarrow x = 27$$ - For $y=10$, find $x$: $$x + 2(10) = 41 \Rightarrow x + 20 = 41 \Rightarrow x = 21$$ - For $x=19$, find $y$: $$19 + 2y = 41 \Rightarrow 2y = 22 \Rightarrow y = 11$$ - For $y=5$, find $x$: $$x + 2(5) = 41 \Rightarrow x + 10 = 41 \Rightarrow x = 31$$ 6. **Complete the table:** | number of chickens (x) | number of cows (y) | |-----------------------|--------------------| | 35 | 3 | | 27 | 7 | | 21 | 10 | | 19 | 11 | | 31 | 5 | **Final answer:** The missing values are $y=3$ for $x=35$, $x=27$ for $y=7$, $x=21$ for $y=10$, $y=11$ for $x=19$, and $x=31$ for $y=5$.