Correctly solve the problem using systems of equations.

Paco and José have geese. José says: if you give me 733 geese, we will have the same amount; Paco answers: if you gave me 733 geese I would have three times more than you had. How many geese does Paco have?

Let's solve this problem using a system of equations. 

### Step 1: Define the Variables
Let:
- \( p \) represent the number of geese Paco has.
- \( j \) represent the number of geese José has.

### Step 2: Create the Equations
From the problem statement, we can form two equations based on the given conditions:

1. **José says:** "If you give me 733 geese, we will have the same amount."
   - This can be written as: 
   \[
   p - 733 = j + 733
   \]

2. **Paco answers:** "If you gave me 733 geese, I would have three times more than you had."
   - This can be written as:
   \[
   p + 733 = 3 \times (j - 733)
   \]

### Step 3: Solve the Equations
Let's solve the system step by step.

#### Equation 1: 
\[
p - 733 = j + 733
\]
Add 733 to both sides:
\[
p - j = 1466 \quad \text{(Equation 1)}
\]

#### Equation 2: 
\[
p + 733 = 3 \times (j - 733)
\]
Expand and simplify:
\[
p + 733 = 3j - 2199
\]
Subtract 733 from both sides:
\[
p = 3j - 2932 \quad \text{(Equation 2)}
\]

### Step 4: Substitute and Solve
Now, we'll substitute Equation 1 into Equation 2.

From Equation 1:
\[
p = j + 1466
\]

Substitute this into Equation 2:
\[
j + 1466 = 3j - 2932
\]
Subtract \( j \) from both sides:
\[
1466 = 2j - 2932
\]
Add 2932 to both sides:
\[
4398 = 2j
\]
Divide by 2:
\[
j = 2199
\]

Now that we have \( j \), we can find \( p \) using Equation 1:
\[
p = 2199 + 1466 = 3665
\]

### Answer:
Paco has **3,665 geese**.

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Would you like more details on this problem or have any questions? Here are some related questions to consider:

1. How would the solution change if the numbers were different?
2. What if Paco had fewer geese than José initially?
3. How can we check the solution to verify it's correct?
4. Can this problem be solved using a different method?
5. What are the implications if the word "three times more" was interpreted differently?
6. How do the variables represent the situation?
7. Can this system be solved graphically? What would the graph look like?
8. How would you explain the steps to someone unfamiliar with systems of equations?

**Tip:** When solving word problems, carefully translate the conditions into equations by paying close attention to the relationships described.

Learn how to solve a word problem involving systems of equations. Paco and José have geese, and the problem involves determining how many geese Paco has based on the conditions given. Suitable for Grades 7-9.

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