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.