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Let's analyze the given problem:

We are told that \( x \) and \( y \) are integers, and \( x = 50y + 69 \). We need to determine which of the following expressions must be odd:

1. \( xy \)
2. \( x + y \)
3. \( x + 2y \)
4. \( 3x - 1 \)
5. \( 3x + 1 \)

### Step 1: Expression for \( x \)
We are given that \( x = 50y + 69 \).

- Since \( 50y \) is always even (multiplying an integer by 50, which is even, results in an even number),
- \( 69 \) is odd.
  
Therefore, \( x = \text{even} + \text{odd} = \text{odd} \).

This tells us that \( x \) is always odd for any integer value of \( y \).

### Step 2: Evaluate Each Expression
1. **\( xy \)**:  
   \( x \) is odd, but we don't know whether \( y \) is odd or even. If \( y \) is even, \( xy \) will be even; if \( y \) is odd, \( xy \) will be odd. Therefore, \( xy \) could be either odd or even.
   
2. **\( x + y \)**:  
   Since \( x \) is odd, the sum \( x + y \) will be odd if \( y \) is even and even if \( y \) is odd. Therefore, this could also be either odd or even depending on \( y \).

3. **\( x + 2y \)**:  
   \( 2y \) is always even (since multiplying any integer by 2 results in an even number), and adding an odd number \( x \) to an even number will always result in an odd number.  
   **Thus, \( x + 2y \) must always be odd.**

4. **\( 3x - 1 \)**:  
   Since \( x \) is odd, \( 3x \) (which is 3 times an odd number) will also be odd. Subtracting 1 from an odd number results in an even number. Therefore, \( 3x - 1 \) will always be even.

5. **\( 3x + 1 \)**:  
   As above, \( 3x \) is odd, and adding 1 to an odd number results in an even number. Thus, \( 3x + 1 \) will also be even.

### Conclusion:
The expression that must be **odd** is **\( x + 2y \)**.

Would you like further clarification on any step?

Here are 5 related questions to explore further:
1. How do you determine whether a sum of an odd and even number is odd or even?
2. What would change in the problem if \( x \) were even instead of odd?
3. How do properties of even and odd numbers behave under multiplication?
4. How can we express other conditions for \( y \) that would change the final answer?
5. Can we generalize this problem for different coefficients like \( 50y \) in \( x = 50y + 69 \)?

**Tip:** Remember, adding or subtracting an even number from an odd number always results in an odd number, and vice versa.

If x and y are integers and x = 50y + 69, which of the following must be odd? (xy, x + y, x + 2y, 3x - 1, 3x + 1)

Solve a problem involving even and odd numbers with the equation x = 50y + 69. Determine which of the given expressions must be odd. Suitable for students in Grades 9-10.

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