In the image, we are given two lines, $\overrightarrow{PR}$ and $\overrightarrow{QS}$, that are perpendicular to each other. To solve for the values of $x$ and $y$, we need to use the fact that the angles formed between these lines must add up to $90^\circ$, since perpendicular lines form a right angle.

From the diagram:

1. Angle $\angle PQT = (4y - 2)^\circ$
2. Angle $\angle SQT = (5x + 6)^\circ$
3. Angle $\angle TQR = 2x^\circ$

Since $\overrightarrow{PR}$ and $\overrightarrow{QS}$ are perpendicular, the sum of the angles along the line $\overrightarrow{PR}$ and $\overrightarrow{QS}$ at point $Q$ must equal $90^\circ$. Specifically:

$(4y - 2)^\circ + (5x + 6)^\circ + 2x^\circ = 90^\circ$

Step 1: Set up the equation

We can combine the terms to form an equation: $(4y - 2) + (5x + 6) + 2x = 90$

Simplify: $4y + 7x + 4 = 90$

Now, subtract 4 from both sides: $4y + 7x = 86$

Step 2: Solve for $x$ and $y$

At this point, we have one equation with two unknowns. However, we may need more information or constraints to find the exact values of $x$ and $y$.

Would you like me to continue solving this equation or provide further assistance based on possible assumptions? Additionally, do you have any clarifications regarding the problem setup?

Here are 5 questions related to the topic:

1. Can you derive similar equations for other geometry problems involving perpendicular lines?
2. How can you verify if two lines are perpendicular using their slopes?
3. What is the geometric significance of the angles at point $Q$?
4. How do linear equations involving multiple variables help in solving geometry problems?
5. How could you generalize this approach to solve more complex geometric configurations?

Tip: When working with angles in geometry, always check if the sum of the angles around a point adds up to $360^\circ$ or if perpendicular lines add up to $90^\circ$.