The image shows a pair of intersecting lines with angles marked as algebraic expressions. Since these angles are formed by intersecting lines, they are vertically opposite angles, meaning they are equal. We can set up two equations based on this principle:

1. $(2y + 6)^\circ = (8y - 102)^\circ$ because they are vertically opposite angles.
2. $3x^\circ = (8y - 102)^\circ$ because they are also vertically opposite angles.

Let’s solve these equations step by step.

1. Solving for $y$:

From the first equation: $(2y + 6) = (8y - 102)$ Simplify and solve for $y$: $2y + 6 = 8y - 102$ $6 + 102 = 8y - 2y$ $108 = 6y$ $y = \frac{108}{6} = 18$

2. Solving for $x$:

Now substitute $y = 18$ into the second equation: $3x = 8(18) - 102$ $3x = 144 - 102$ $3x = 42$ $x = \frac{42}{3} = 14$

Thus, the values of $y$ and $x$ are: $x = 14, \quad y = 18$

Would you like further clarification or details on any of these steps?

Here are some related questions to explore further:

1. What is the geometric significance of vertically opposite angles?
2. Can you explain the steps to solve a system of linear equations?
3. How do intersecting lines form supplementary angles?
4. What would change if the lines were parallel instead of intersecting?
5. How are angle measurements used in real-life applications?

Tip: When solving for variables in geometry, always check if the angles formed are supplementary or vertically opposite, as this helps set up your equations accurately.