https://file.aiquickdraw.com/mathbot/file/1726028708407n36lewef.jpg

In the diagram, we see that the angles formed are vertical angles, which means opposite angles are equal.

From the diagram:

- \( (4x - y)^\circ \) and \( 5y^\circ \) are opposite angles, so:
  \[
  4x - y = 5y
  \]

- \( (x + y)^\circ \) is also a vertical angle, so it equals \( 5y^\circ \). Thus:
  \[
  x + y = 5y
  \]

### Step 1: Solve the system of equations
We now have two equations:
1. \( 4x - y = 5y \)
2. \( x + y = 5y \)

### Step 2: Solve for \( y \)
From the second equation \( x + y = 5y \), solve for \( x \) in terms of \( y \):
\[
x = 5y - y = 4y
\]

### Step 3: Substitute \( x = 4y \) into the first equation
Now substitute \( x = 4y \) into the first equation \( 4x - y = 5y \):
\[
4(4y) - y = 5y
\]
\[
16y - y = 5y
\]
\[
15y = 5y
\]
Subtract \( 5y \) from both sides:
\[
10y = 0
\]
Thus:
\[
y = 0
\]

### Step 4: Find \( x \)
Substitute \( y = 0 \) back into \( x = 4y \):
\[
x = 4(0) = 0
\]

### Conclusion
Both \( x \) and \( y \) are 0. Since \( x = 0 \) and \( y = 0 \), all the angles in the diagram will also measure 0°. However, this result might indicate a need for rechecking the question context or parameters, as having all variables and angles equal to zero could be unusual in this context.

Would you like further clarification or details?

Here are some related questions to explore further:
1. What are vertical angles, and why are they equal?
2. How do you solve systems of equations with two variables?
3. Why might the result of \( x = 0 \) and \( y = 0 \) suggest a potential issue?
4. Can geometric problems have trivial or zero solutions?
5. How do linear equations behave when multiple variables are involved?

**Tip**: Always check your solution back in the original problem to verify that the result makes sense within the context of the problem.

Find the value of each variable and the measure of each angle from the given equations (4x - y)°, 5y°, and (x + y)°.

Solve the system of equations derived from angle properties to find the values of x and y in (4x - y)°, 5y°, and (x + y)°. Learn key concepts in algebra and geometry, suitable for students in Grades 8-10.

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