Given the problem, we need to solve for $x$ and $y$ using the provided angles and equations. Here are the equations given:

1. $\angle XET = 90^\circ$
2. $\angle XES + \angle SEP + \angle PET = 90^\circ$
3. $\angle SEP = 5x - 4y + 2$
4. $\angle PET = 3x + 2y$

Step 1: Substitute the given angles into the second equation

The second equation states that: $\angle SEP + \angle PET = 90^\circ - \angle XES$ But since $\angle XES + \angle SEP + \angle PET = 90^\circ$, we can directly set up: $(5x - 4y + 2) + (3x + 2y) = 90^\circ$

Step 2: Combine like terms

Let's simplify the equation: $5x - 4y + 2 + 3x + 2y = 90^\circ$ $8x - 2y + 2 = 90$

Step 3: Solve for $x$ and $y$

Now, we'll solve the simplified equation: $8x - 2y = 90 - 2$ $8x - 2y = 88$ Divide the entire equation by 2: $4x - y = 44 \quad \text{(Equation 1)}$

Now, to solve for $x$ and $y$, we'll need another equation. We can directly use the angles given:

Step 4: Use angle $\angle XET = 90^\circ$

Since $\angle XET = 90^\circ$ and no further details are provided linking this with other equations, we focus on Equation 1.

Let's assume for now there's no other equation provided and solve for one variable:

$y = 4x - 44 \quad \text{(substituting into Equation 1)}$

Step 5: Conclusion

Without additional constraints or relationships, $y = 4x - 44$ is the simplest form of solution given the available information. However, to find specific values for $x$ and $y$, more details or an additional equation would be required.

Do you need more details or have any questions?

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Here are 5 related questions you might find interesting:

1. How would the solution change if $\angle XES$ was given?
2. What happens to the equation if $\angle SEP$ or $\angle PET$ is 0?
3. Can you solve for $x$ and $y$ using another method, like substitution or elimination, if more equations were provided?
4. How does changing one angle impact the others in a geometric configuration?
5. What additional information would help to find exact values for $x$ and $y$?

Tip: When solving for multiple variables, always check if you have enough independent equations to uniquely determine each variable.