It looks like you're trying to solve the following system of linear equations:

1. \( 3x - 2y = 5 \)
2. \( -6x + 4y = -10 \)

Let's solve this system step by step using either substitution or elimination. I'll proceed with the elimination method for clarity.

### Step 1: Multiply the first equation to eliminate the variable \( y \)

To eliminate \( y \), notice that the second equation has a term \( 4y \), which is double the term \( -2y \) in the first equation. Let's multiply the first equation by 2 to align the coefficients of \( y \):

\[
2(3x - 2y) = 2(5)
\]
This simplifies to:
\[
6x - 4y = 10
\]

Now we have:
1. \( 6x - 4y = 10 \)
2. \( -6x + 4y = -10 \)

### Step 2: Add the two equations together

Now, we add the two equations together to eliminate \( y \):

\[
(6x - 4y) + (-6x + 4y) = 10 + (-10)
\]

This simplifies to:

\[
0 = 0
\]

### Step 3: Interpret the result

The equation \( 0 = 0 \) indicates that the two equations are not independent, meaning they represent the same line. Therefore, the system has infinitely many solutions.

### Conclusion

The system of equations represents the same line, so there are **infinitely many solutions**. Any pair \( (x, y) \) that satisfies one equation will also satisfy the other.

Would you like more details on finding the parametric form of the solutions, or do you have any other questions?

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Here are 5 related questions for further exploration:
1. How do you determine if a system of equations has no solution?
2. What is the difference between dependent and independent systems of equations?
3. Can you explain what it means when two lines are coincident?
4. How can we graphically interpret the solution to a system of linear equations?
5. How do we solve a system of equations using substitution?

**Tip:** When solving systems of linear equations, look at the coefficients to identify patterns, like multiples or opposites, which can simplify the process!

Explore the step-by-step solution to solve the system of linear equations 3x - 2y = 5 and -6x + 4y = -10 using the elimination method. Understand key algebraic concepts like dependent systems and how to recognize when equations represent the same line. Suitable for students in Grades 8-10.

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