To solve this system by elimination, let's examine the equations given:

1. $4x + 3y = 1$
2. $-12x - 9y = -3$

Step 1: Simplify the System

Notice that the second equation is a multiple of the first equation. If we multiply the first equation by $-3$, we get:

$-3(4x + 3y) = -3 \cdot 1$ which simplifies to:

$-12x - 9y = -3$

This is identical to the second equation. Therefore, these two equations are dependent (they represent the same line).

Conclusion

Since the equations are dependent, there are infinitely many solutions. The solution set represents all points that lie on the line defined by either equation.

So, the correct answer is: B. There are infinitely many solutions, and the equations are dependent. The solution set is $\{(x, y) \mid 4x + 3y = 1\}$.

Would you like further details on how to interpret dependent equations or have any other questions?

Here are 5 additional questions to deepen your understanding:

1. How can you tell if two equations in a system are dependent without using elimination?
2. What does it mean geometrically when a system has infinitely many solutions?
3. How would you modify one of the equations to create a system with a single solution instead?
4. What is the graphical interpretation of an inconsistent system of equations?
5. How does elimination differ from substitution when solving systems of equations?

Tip: When checking for dependent equations, look for scalar multiples between the equations.