https://file.aiquickdraw.com/mathbot/file/1728237324721etnm6t3x.jpg

The system of linear equations given in the image is:

\[
\text{(1)} \ \ 6x - 5y = 0
\]
\[
\text{(2)} \ \ 12x - 10y = -3
\]

We are asked to solve this system by the method of elimination.

### Step 1: Analyze and simplify the equations

Looking at the two equations, we can notice that equation (2) is just a multiple of equation (1). Specifically, equation (2) is twice equation (1), except for the constant term on the right-hand side.

If we multiply equation (1) by 2, we get:
\[
2(6x - 5y) = 2(0)
\]
\[
12x - 10y = 0
\]

This new equation, \[12x - 10y = 0\], is clearly inconsistent with the original equation (2), which is \[12x - 10y = -3\].

### Step 2: Interpret the result

Since multiplying equation (1) by 2 gives us \[12x - 10y = 0\], but equation (2) is \[12x - 10y = -3\], we see that there is a contradiction. This means that the system of equations has **no solution**. The lines represented by these equations are parallel and never intersect.

### Conclusion:
The solution set for this system of linear equations is **empty** because there is no solution.

Would you like further clarification on any step or additional details?

Here are some related questions:
1. How can you graphically represent a system with no solution?
2. What does it mean geometrically when a system of linear equations has no solution?
3. Can a system with inconsistent equations be solved by substitution instead of elimination?
4. How do you recognize parallel lines in a system of linear equations?
5. What is the difference between inconsistent and dependent systems of equations?

**Tip**: Always check for proportionality between the coefficients of the variables to quickly identify whether a system might be inconsistent.

Determine the solution set of the system of linear equations by elimination. 6x - 5y = 0 and 12x - 10y = -3.

Learn how to determine that a system of linear equations has no solution using the elimination method. We explore a system where multiplying one equation results in a contradiction, indicating parallel lines. Suitable for students in Grades 8-10.

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