1. **State the problem:** There are 18 animals in total, some chickens and some cows. The total number of legs counted is 50. We need to find how many chickens and how many cows there are. 2. **Define variables:** Let $x$ be the number of chickens and $y$ be the number of cows. 3. **Write equations based on the problem:** - Total animals: $$x + y = 18$$ - Total legs: Chickens have 2 legs each, cows have 4 legs each, so $$2x + 4y = 50$$ 4. **Solve the system of equations:** From the first equation, express $y$ in terms of $x$: $$y = 18 - x$$ Substitute into the second equation: $$2x + 4(18 - x) = 50$$ 5. **Simplify and solve for $x$:** $$2x + 72 - 4x = 50$$ $$\cancel{2x} + 72 - \cancel{4x} = 50$$ becomes $$-2x + 72 = 50$$ Subtract 72 from both sides: $$-2x = 50 - 72$$ $$-2x = -22$$ Divide both sides by -2: $$\frac{-2x}{\cancel{-2}} = \frac{-22}{\cancel{-2}}$$ $$x = 11$$ 6. **Find $y$:** $$y = 18 - x = 18 - 11 = 7$$ 7. **Answer:** There are 11 chickens and 7 cows in the barnyard.