https://file.aiquickdraw.com/mathbot/file/1725843230586yzzx0xov.jpg

The problem states:

"A farmer notes that in a field full of horses and geese there are 26 heads and 86 feet. How many of each animal are there?"

### Let’s define:
- \( h \) = number of horses
- \( g \) = number of geese

### We know:
1. Each animal has 1 head:  
   - \( h + g = 26 \)  (since there are 26 heads)
2. Horses have 4 feet, and geese have 2 feet:  
   - \( 4h + 2g = 86 \) (since there are 86 feet)

### Solving the System of Equations:

1. From the first equation:  
   \( h + g = 26 \)  
   Solve for \( g \):  
   \( g = 26 - h \)

2. Substitute \( g = 26 - h \) into the second equation:  
   \( 4h + 2(26 - h) = 86 \)

3. Expand and simplify:  
   \( 4h + 52 - 2h = 86 \)  
   \( 2h + 52 = 86 \)  
   \( 2h = 86 - 52 \)  
   \( 2h = 34 \)  
   \( h = 17 \)

4. Now substitute \( h = 17 \) back into the equation for \( g \):  
   \( g = 26 - 17 = 9 \)

### Answer:
- There are 17 horses and 9 geese.

Would you like more details or have any questions?

### Related Questions:
1. How would the problem change if the animals had different numbers of legs?
2. What if there were 30 heads and 100 feet? How would the solution change?
3. How can you verify the solution using another method, such as substitution or elimination?
4. How would the system of equations change if one of the animals had missing legs?
5. Can you represent this problem graphically?

**Tip:** For systems of linear equations, substitution is a great method when one variable can be easily expressed in terms of the other.

A farmer notes that in a field full of horses and geese there are 26 heads and 86 feet. How many of each animal are there?

Learn how to solve a system of linear equations to determine the number of horses and geese in a field, given 26 heads and 86 feet. This problem involves key algebraic concepts such as substitution and solving linear equations, suitable for students in Grades 6-8.

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