The given system of equations is:

$-6x + 4y = 2$ $3x - 2y = -1$

To determine how many $(x, y)$ solutions this system has, we will analyze the equations by solving them.

Step 1: Multiply the second equation by 2.

$2 \times (3x - 2y) = 2 \times (-1)$ This gives: $6x - 4y = -2$

Step 2: Add this new equation to the first equation.

$-6x + 4y + (6x - 4y) = 2 + (-2)$ Simplifying: $0 = 0$

This result indicates that the two equations are dependent, meaning they represent the same line. Therefore, there are infinitely many solutions to this system, as every point on the line satisfies both equations.

Conclusion:

The system has infinitely many solutions.

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you graphically represent a system with infinitely many solutions?
2. What does it mean when two equations are dependent in a system of linear equations?
3. How can substitution or elimination methods show the same results for dependent systems?
4. What changes in the system of equations would lead to exactly one solution?
5. How can you verify if two equations are multiples of each other?

Tip: When two linear equations simplify to an identity (like $0 = 0$), the system has infinitely many solutions because the equations represent the same line.