Directions: Solve for x and y.

angle XET = 90 
angle XES + angle SEP + angle PET = 90
angle SEP = 5x - 4y + 2 
angle PET = 3x + 2y

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.

Directions: Solve for x and y.

angle XET = 90
direction XES + angle SEP + angle PET = 90
angle SEP = 5x - 4y + 2
angle PET = 3x + 2y

Learn how to solve for x and y using given angle relationships in geometry. This problem involves algebraic expressions and the angle sum property of a right angle. Suitable for students in Grades 8-10.

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