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We are given one equation in a system of linear equations:

\[
-15x + 25y = 65
\]

The problem states that the system has infinitely many solutions. For this to occur, the two equations must be scalar multiples of each other, meaning that the second equation must represent the same line (it can be a multiple or rearranged version of the first equation).

### Step-by-Step Analysis:

1. Simplify the given equation:

\[
-15x + 25y = 65
\]

Dividing the entire equation by 5:

\[
-3x + 5y = 13
\]

Now, let's check which of the answer choices matches this equation or a multiple of this equation.

2. Compare the coefficients in each of the given choices:

- Option A: \(12x + 20y = 52\)
    - Divide by 4: \(3x + 5y = 13\)
    - This is not the same equation as \(-3x + 5y = 13\) (the sign of \(x\) is different).
  
- Option B: \(12x + 20y = -52\)
    - Divide by 4: \(3x + 5y = -13\)
    - This also doesn't match.

- Option C: \(-12x + 20y = 52\)
    - Divide by \(-4\): \(3x - 5y = -13\)
    - This doesn't match either.

- Option D: \(-12x + 20y = -52\)
    - Divide by \(-4\): \(3x - 5y = 13\)
    - This matches the simplified form of the original equation \(-3x + 5y = 13\).

### Conclusion:
The second equation must be a scalar multiple of the first one. Hence, the correct answer is **D**.

Would you like further clarification on any part of this solution?

Here are some related questions you might find useful:
1. What conditions ensure a system of two linear equations has infinitely many solutions?
2. How can you determine if two lines are parallel or identical based on their equations?
3. What happens when two linear equations in a system have no solutions?
4. How can we solve a system of linear equations algebraically?
5. How can you verify if two lines are coincident geometrically?

**Tip**: When checking if two lines are identical, always simplify both equations and compare their coefficients. If they are proportional, the lines are the same.

One of the two equations in a system of linear equations is given: -15x + 25y = 65. The system has infinitely many solutions. Which of the following could be the second equation in the system? A) 12x + 20y = 52 B) 12x + 20y = -52 C) -12x + 20y = 52 D) -12x + 20y = -52

Explore how to determine the second equation in a system of linear equations that has infinitely many solutions. Learn the step-by-step process to identify scalar multiples of equations, including simplification techniques. Suitable for high school students in Grades 9-11.

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