There is an animal farm where chickens and cows live. All together, there are 71 heads and 216 legs. How many chickens and cows are there on the farm?

chickens   and     cows

Let's define two variables:

- Let \( x \) be the number of chickens.
- Let \( y \) be the number of cows.

From the problem, we know:

1. Each chicken has 1 head and each cow also has 1 head. Therefore, the total number of heads is:
   \[
   x + y = 71
   \]
   
2. Each chicken has 2 legs and each cow has 4 legs. Therefore, the total number of legs is:
   \[
   2x + 4y = 216
   \]

We now have the system of equations:
\[
x + y = 71
\]
\[
2x + 4y = 216
\]

### Solving the system of equations:

First, solve the first equation for \( y \):
\[
y = 71 - x
\]

Substitute \( y = 71 - x \) into the second equation:
\[
2x + 4(71 - x) = 216
\]

Simplify and solve for \( x \):
\[
2x + 284 - 4x = 216
\]
\[
-2x + 284 = 216
\]
\[
-2x = 216 - 284
\]
\[
-2x = -68
\]
\[
x = \frac{-68}{-2} = 34
\]

So, there are \( 34 \) chickens.

Now, substitute \( x = 34 \) into the first equation to find \( y \):
\[
34 + y = 71
\]
\[
y = 71 - 34 = 37
\]

Thus, there are \( 37 \) cows.

### Final Answer:
- Number of chickens: \( 34 \)
- Number of cows: \( 37 \)

Would you like more details or have any questions? 

Here are some related questions:
1. What if the total number of heads and legs were different?
2. How would this problem change if some animals were added?
3. Can we solve this problem using matrices or another method?
4. How can we graph the system of equations?
5. What is the significance of solving systems of linear equations in real life?

**Tip:** Always double-check by substituting your values back into the original equations to verify the solution.

There is an animal farm where chickens and cows live. All together, there are 71 heads and 216 legs. How many chickens and cows are there on the farm?

Learn how to solve a classic algebra problem involving chickens and cows on a farm with 71 heads and 216 legs. This step-by-step solution demonstrates using systems of linear equations and the substitution method to find the number of chickens and cows. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation