We are given two equations:

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

To find the $y$-coordinate of their intersection, we can solve these simultaneously.

Step 1: Add the two equations to eliminate $x$.

From equation (1): $2y - x = 3 \quad \Rightarrow \quad -x = 3 - 2y \quad \Rightarrow \quad x = 2y - 3$

Substitute this expression for $x$ into equation (2):

$3y + (2y - 3) = -4$

Step 2: Simplify and solve for $y$.

$3y + 2y - 3 = -4$ $5y - 3 = -4$ $5y = -4 + 3$ $5y = -1$ $y = -\frac{1}{5}$

Thus, the $y$-coordinate of the intersection is $y = -\frac{1}{5}$.

The correct answer is $\boxed{-\frac{1}{5}}$, which corresponds to option C.

Would you like further details on this method or any other aspect of the problem?

Here are five related questions to explore:

1. How can we verify the solution for the intersection point?
2. What is the significance of solving systems of linear equations geometrically?
3. Can substitution work better than elimination for other types of systems?
4. How do parallel lines behave in systems of equations?
5. What is the general form for solving 3-variable systems?

Tip: For systems of linear equations, the elimination method is usually faster when one variable has coefficients that can easily cancel out.