We are given the following system of linear equations:

1. $6x - 4 = -9y$
2. $-12x - 18y = -8$

Step 1: Simplify the first equation

Rearranging the first equation for easier substitution: $6x - 4 = -9y \quad \Rightarrow \quad 6x + 9y = 4 \quad \text{(Equation 1 simplified)}$

Step 2: Rewrite the second equation

The second equation is already simplified: $-12x - 18y = -8 \quad \text{(Equation 2)}$

Step 3: Analyze the system

The two equations we have now are:

1. $6x + 9y = 4$
2. $-12x - 18y = -8$

Notice that the second equation is simply a multiple of the first equation. If we multiply the first equation by $-2$, we get:

$-2(6x + 9y) = -2(4) \quad \Rightarrow \quad -12x - 18y = -8$

This matches the second equation exactly. Therefore, the two equations are dependent.

Conclusion:

Since the two equations are dependent (essentially the same line), the system has infinitely many solutions.

Would you like more details or further explanation? Here are some related questions to deepen your understanding:

1. What does it mean for a system of equations to have infinitely many solutions?
2. How can you tell if two equations are dependent in general?
3. What is the geometric interpretation of a system with infinitely many solutions?
4. How do you solve a system of equations using substitution or elimination when there are infinitely many solutions?
5. Can a system of linear equations have no solutions, and how can we identify such cases?

Tip: A system of equations will have infinitely many solutions when the two equations represent the same line (i.e., one equation is a scalar multiple of the other).