In this image, we are looking at two pairs of vertically opposite angles.

Given that the vertical angles are equal:

1. The angles labeled $(6x - 11)^\circ$ and $(4x + 23)^\circ$ are vertical angles.
2. The angles labeled $(9y - 19)^\circ$ and $(4x + 23)^\circ$ are also vertical angles.

Step 1: Solve for $x$

We first equate the two expressions for the vertical angles involving $x$: $6x - 11 = 4x + 23$ Solve for $x$: $6x - 4x = 23 + 11$ $2x = 34$ $x = 17$

Step 2: Use $x$ to find $y$

Now, equate the other pair of vertical angles, involving $y$: $9y - 19 = 4x + 23$ Substitute $x = 17$: $9y - 19 = 4(17) + 23$ $9y - 19 = 68 + 23$ $9y - 19 = 91$ Solve for $y$: $9y = 91 + 19$ $9y = 110$ $y = \frac{110}{9} \approx 12.22$

Thus, the value of $y$ is approximately $12.22$.

Let me know if you want any more details or further clarifications!

Related Questions:

1. What are vertically opposite angles, and why are they equal?
2. How would this solution change if the angles were supplementary instead of vertical?
3. Can you find the measure of each angle now that we know $x$ and $y$?
4. What is the geometric significance of vertical angles in parallel line intersections?
5. How do you check if an angle is acute, obtuse, or right based on algebraic expressions like these?

Tip: Always check that your equations are set up correctly by verifying which angles are equal or supplementary, depending on the problem.