What if there were 120 legs and 40 heads? How many sheep and ducks?

Let's solve this problem similarly to the previous one.

### Step 1: Define the Variables
- Let \( x \) be the number of sheep.
- Let \( y \) be the number of ducks.

### Step 2: Set Up the Equations
1. **Heads equation**: The total number of heads is given by:
   \[
   x + y = 40
   \]
2. **Legs equation**: Sheep have 4 legs and ducks have 2 legs, so the total number of legs is given by:
   \[
   4x + 2y = 120
   \]

### Step 3: Solve the System of Equations
1. **Simplify the legs equation** by dividing it by 2:
   \[
   2x + y = 60
   \]

2. **Subtract the heads equation from the simplified legs equation**:
   \[
   (2x + y) - (x + y) = 60 - 40
   \]
   Simplifying this, we get:
   \[
   x = 20
   \]

3. **Substitute \( x = 20 \) into the heads equation**:
   \[
   20 + y = 40
   \]
   Solving for \( y \), we get:
   \[
   y = 20
   \]

### Conclusion
The farmer has **20 sheep** and **20 ducks**.

Would you like any further explanations or do you have any more questions?

Here are 5 related questions to explore further:
1. What if there were 80 legs and 30 heads? How many sheep and ducks?
2. How does changing the number of animals affect the number of legs if the ratio of sheep to ducks is fixed?
3. Can this method be used to solve problems involving more than two types of animals?
4. How would the equations change if one animal had 3 legs instead of 2 or 4?
5. What is the general strategy for solving systems of linear equations in word problems?

**Tip**: Always simplify the equations when possible to make the system easier to solve.

Learn how to solve a word problem involving 120 legs and 40 heads to determine the number of sheep and ducks. Step-by-step solution provided using systems of linear equations. Suitable for Grades 7-9 students.

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