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Let's analyze the given sets \( A \) and \( B \) to find their intersection \( A \cap B \).

The sets are defined as:

- \( A = \{(x, y) \mid x + y = 7, x > y, x, y \in \mathbb{R} \} \)
- \( B = \{(x, y) \mid x - y = 3, x > y, x, y \in \mathbb{R} \} \)

### Step 1: Analyze Set \( A \)
For \( A \), the condition is:
\[
x + y = 7
\]
This can be rewritten to express \( y \) in terms of \( x \):
\[
y = 7 - x
\]
We also have the condition \( x > y \), so:
\[
x > 7 - x
\]
Solving this inequality:
\[
2x > 7 \quad \Rightarrow \quad x > \frac{7}{2} = 3.5
\]
Thus, for \( (x, y) \in A \), \( x > 3.5 \) and \( y = 7 - x \).

### Step 2: Analyze Set \( B \)
For \( B \), the condition is:
\[
x - y = 3
\]
This can be rewritten to express \( y \) in terms of \( x \):
\[
y = x - 3
\]
We also have the condition \( x > y \), so:
\[
x > x - 3
\]
This inequality is always true, so no additional restriction on \( x \) is needed.

### Step 3: Find the Intersection \( A \cap B \)
The intersection \( A \cap B \) consists of pairs \( (x, y) \) that satisfy both sets' conditions. This means:
\[
x + y = 7 \quad \text{and} \quad x - y = 3
\]

We can solve these two equations simultaneously:

1. Add the two equations:
\[
(x + y) + (x - y) = 7 + 3 \quad \Rightarrow \quad 2x = 10 \quad \Rightarrow \quad x = 5
\]

2. Substitute \( x = 5 \) into either equation (let's use \( x + y = 7 \)):
\[
5 + y = 7 \quad \Rightarrow \quad y = 2
\]

So, the only solution that satisfies both sets is \( (x, y) = (5, 2) \).

### Conclusion
The intersection \( A \cap B \) is the singleton set:
\[
A \cap B = \{(5, 2)\}
\]

Would you like to know more details, or have any questions?

Here are five related questions to expand on this:

1. What would happen if the conditions \( x > y \) were removed from both sets?
2. How would the intersection change if the set \( B \) was defined with a different linear equation, such as \( x - y = 2 \)?
3. Can you find the union of sets \( A \) and \( B \) instead of their intersection?
4. What would the intersection \( A \cap B \) be if \( A \) and \( B \) were defined in three-dimensional space?
5. How would the problem change if \( x \) and \( y \) were restricted to integers?

**Tip:** When solving systems of linear equations, it's helpful to add or subtract the equations to eliminate one variable, making it easier to find the solution.

A = {(x, y) | x + y = 7, x > y, x, y ∈ ℝ} B = {(x, y) | x - y = 3, x > y, x, y ∈ ℝ} Find A ∩ B

Solve the system of equations x + y = 7 and x - y = 3 to find the intersection of two sets A and B. Learn how to solve linear systems and inequalities. Suitable for students in Grades 10-12.

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