To determine whether $(-3, -3)$ is a solution to the given system of equations, we substitute $x = -3$ and $y = -3$ into both equations and check if both equations hold true.

The system of equations is:

7x - 6y = -3 \tag{1} 13x - 12y = -3 \tag{2}

Step 1: Substitute $(-3, -3)$ into Equation (1)

$7(-3) - 6(-3) = -3$

Simplify:

$-21 + 18 = -3$

$-3 = -3$

This is true, so $(-3, -3)$ satisfies Equation (1).

Step 2: Substitute $(-3, -3)$ into Equation (2)

$13(-3) - 12(-3) = -3$

Simplify:

$-39 + 36 = -3$

$-3 = -3$

This is also true, so $(-3, -3)$ satisfies Equation (2).

Conclusion:

Since $(-3, -3)$ satisfies both equations, it is a solution to the system of equations.

Would you like any further explanation or additional details? Here are a few related questions to deepen your understanding:

1. What happens if you substitute a different pair of values into this system of equations?
2. How can you graph the system of equations to visualize the solution?
3. What are the methods to solve a system of equations algebraically (substitution, elimination)?
4. How do you determine if a system has one solution, no solution, or infinitely many solutions?
5. Can this system of equations be represented as a matrix equation? If so, how?

Tip: Always check both equations when substituting a proposed solution to ensure it satisfies the entire system.