In each of the following, state which of the ordered pairs belong to the given relations? `{(x,y):y=(x+3)/(x-3),xne1}:(0,1),(2,5),(5,2),(3,3),(7,5),(7,(5)/(3))`

Answer

To determine which ordered pairs belong to the relation defined by \( y = \frac{x + 3}{x - 3} \) and \( x \neq 1 \), we will evaluate each ordered pair one by one.

### Step-by-Step Solution:

1. **Evaluate the first ordered pair (0, 1)**:
- Here, \( x = 0 \).
- Calculate \( y \):
\[ y = \frac{0 + 3}{0 - 3} = \frac{3}{-3} = -1 \]
- The ordered pair (0, 1) has \( y = 1 \), but we calculated \( y = -1 \).
- **Conclusion**: (0, 1) does not belong to the relation.

2. **Evaluate the second ordered pair (2, 5)**:
- Here, \( x = 2 \).
- Calculate \( y \):
\[ y = \frac{2 + 3}{2 - 3} = \frac{5}{-1} = -5 \]
- The ordered pair (2, 5) has \( y = 5 \), but we calculated \( y = -5 \).
- **Conclusion**: (2, 5) does not belong to the relation.

3. **Evaluate the third ordered pair (5, 2)**:
- Here, \( x = 5 \).
- Calculate \( y \):
\[ y = \frac{5 + 3}{5 - 3} = \frac{8}{2} = 4 \]
- The ordered pair (5, 2) has \( y = 2 \), but we calculated \( y = 4 \).
- **Conclusion**: (5, 2) does not belong to the relation.

4. **Evaluate the fourth ordered pair (3, 3)**:
- Here, \( x = 3 \).
- Calculate \( y \):
\[ y = \frac{3 + 3}{3 - 3} = \frac{6}{0} \]
- The value of \( y \) is undefined (infinity), while the ordered pair (3, 3) has \( y = 3 \).
- **Conclusion**: (3, 3) does not belong to the relation.

5. **Evaluate the fifth ordered pair (7, 5)**:
- Here, \( x = 7 \).
- Calculate \( y \):
\[ y = \frac{7 + 3}{7 - 3} = \frac{10}{4} = \frac{5}{2} \]
- The ordered pair (7, 5) has \( y = 5 \), but we calculated \( y = \frac{5}{2} \).
- **Conclusion**: (7, 5) does not belong to the relation.

6. **Evaluate the sixth ordered pair (7, \frac{5}{3})**:
- Here, \( x = 7 \).
- Calculate \( y \):
\[ y = \frac{7 + 3}{7 - 3} = \frac{10}{4} = \frac{5}{2} \]
- The ordered pair (7, \frac{5}{3}) has \( y = \frac{5}{3} \), but we calculated \( y = \frac{5}{2} \).
- **Conclusion**: (7, \frac{5}{3}) does not belong to the relation.

### Final Conclusion:
None of the ordered pairs (0, 1), (2, 5), (5, 2), (3, 3), (7, 5), or (7, \frac{5}{3}) belong to the given relation.