Let's solve this system of equations step-by-step.

The given system of equations is:

$y = \frac{1}{3}x + 4 \quad \text{(1)}$ $3y - x = 12 \quad \text{(2)}$

Substitution Method:

1. From equation (1), we already have an expression for $y$. Now, we can substitute this value of $y$ into equation (2).

Substitute $y = \frac{1}{3}x + 4$ into equation (2): $3\left(\frac{1}{3}x + 4\right) - x = 12$

2. Simplify the left side: $x + 12 - x = 12$

3. After simplification: $12 = 12$

This equation is true for all values of $x$, meaning that the system of equations is dependent, and the two equations represent the same line. Therefore, there are infinitely many solutions.

Conclusion:

The correct answer is: There are infinitely many solutions.

Would you like more details or have any questions? Here are 5 related questions:

1. What does it mean for a system of equations to have infinitely many solutions?
2. How can you tell if a system of equations is dependent or independent?
3. What is the geometric interpretation of dependent equations?
4. Can substitution always be used to solve a system of equations? Why or why not?
5. How would the solution change if the system had no solutions?

Tip: When solving systems of equations, always check if they are consistent, inconsistent, or dependent by analyzing their graphical relationships or using substitution/elimination methods.