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We are given the problem where there are goats and chickens on a farm, and we need to find how many of each there are. The information provided tells us that:

- Each goat has 1 head and 4 legs.
- Each chicken has 1 head and 2 legs.
- The total number of heads is 8, and the total number of legs is 22.

We can set up a system of equations to solve this.

Let:
- \( x \) represent the number of goats.
- \( y \) represent the number of chickens.

From the information:
1. Since each animal has 1 head, the total heads equation is:
   \[
   x + y = 8
   \]
2. For the legs, since goats have 4 legs and chickens have 2 legs, the total legs equation is:
   \[
   4x + 2y = 22
   \]

Now, we can solve these equations step by step.

### Step 1: Simplify the legs equation
Divide the legs equation by 2 to simplify:
\[
2x + y = 11
\]

### Step 2: Solve the system of equations
We now have:
1. \( x + y = 8 \)
2. \( 2x + y = 11 \)

Subtract the first equation from the second:
\[
(2x + y) - (x + y) = 11 - 8
\]
\[
x = 3
\]

### Step 3: Find \( y \)
Now substitute \( x = 3 \) into the first equation:
\[
3 + y = 8
\]
\[
y = 5
\]

### Final Answer:
- Number of goats = 3
- Number of chickens = 5

Let me know if you need more details or have any questions. Here are some related questions to deepen your understanding:

1. How would the equations change if chickens had 3 legs each?
2. What if there were 10 heads and 26 legs? How many goats and chickens would there be?
3. Could you solve this using a matrix? How?
4. How does the number of legs per animal affect the solution process?
5. How would you solve this problem graphically?

Tip: For systems of equations, elimination and substitution methods are powerful tools for finding solutions.

A mixed farm has some goats and chickens. The goats have 4 legs and the chickens have 2 legs. Alex counted the heads and the legs. The total was 8 heads and 22 legs. How many goats and chickens are on the farm?

Learn how to use systems of linear equations to determine the number of goats and chickens on a farm, based on the counts of heads and legs. This problem demonstrates the application of algebra to solve real-life scenarios, suitable for students in Grades 6-8.

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