The image shows a system of intersecting lines with angle expressions involving variables $x$ and $y$. We are tasked with finding the values of $x$ and $y$.

Here's the breakdown of the angles given:

• One angle is $(18y + 5)^\circ$,
• Another angle is $(10x - 61)^\circ$,
• A third angle is $(x + 10)^\circ$,
• A fourth angle is directly across from $(18y + 5)^\circ$, and they are vertical angles.

Step 1: Set up the vertical angle equation

Since vertical angles are equal, we can set the following equation:

$18y + 5 = 10x - 61$

Step 2: Use the linear pair property

Two adjacent angles on a straight line sum to $180^\circ$. Therefore, the angles $(x + 10)^\circ$ and $(10x - 61)^\circ$ form a linear pair, so:

$(x + 10) + (10x - 61) = 180$

Now let's solve these equations to find $x$ and $y$. I'll work on the algebra next.The values of the variables are:

• $x = 21$
• $y = 8$

Would you like a further explanation on how these values were derived, or do you have any other questions?

Here are some related questions to further expand on this topic:

1. What are vertical angles, and why are they equal?
2. How can you recognize linear pairs of angles in other geometrical figures?
3. What properties of angles do intersecting lines always exhibit?
4. How would you verify that these values of $x$ and $y$ satisfy both angle equations?
5. What other methods could you use to solve systems of equations in geometry?

Tip: In geometry problems, always look for properties like vertical angles or supplementary angles to set up equations and solve for unknowns.