The problem provided asks to find the values of $x$ and $y$ based on a system of equations created from the geometry diagram. The system appears to relate parallel lines with transversal angles, giving rise to algebraic expressions for $x$ and $y$.

From the image, the corresponding angles formed between the parallel lines and transversals must be equal. The equations for the angles are as follows:

1. The first pair of angles gives us: $20 - 5x = y$

2. The second pair of angles gives: $2x + 6 = \frac{3}{5}y + 2$

Now, we will solve these two equations step-by-step.

Step 1: Solve for $y$ from the first equation:

$y = 20 - 5x$

Step 2: Substitute $y = 20 - 5x$ into the second equation:

$2x + 6 = \frac{3}{5}(20 - 5x) + 2$

Step 3: Simplify the equation:

$2x + 6 = \frac{3}{5}(20 - 5x) + 2$ $2x + 6 = \frac{3}{5} \times 20 - \frac{3}{5} \times 5x + 2$ $2x + 6 = 12 - 3x + 2$ $2x + 6 = 14 - 3x$

Step 4: Solve for $x$:

Add $3x$ to both sides: $5x + 6 = 14$ Subtract 6 from both sides: $5x = 8$ Now, divide by 5: $x = \frac{8}{5} = 1.6$

Step 5: Substitute $x = 1.6$ back into the equation for $y$:

$y = 20 - 5(1.6)$ $y = 20 - 8 = 12$

Thus, the solutions are: $x = 1.6, \quad y = 12$

Let me know if you would like further details or have any questions!

Here are 5 related questions to expand on this:

1. What are corresponding angles in parallel lines cut by a transversal?
2. How do you solve a system of equations with substitution?
3. What happens when the system of equations has no solution?
4. Can you solve the system of equations using the elimination method?
5. How can these types of problems appear in real-world applications?

Tip: When solving systems of equations, it's often useful to substitute one variable and simplify step-by-step to avoid mistakes.