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The diagram shows angles around point \( F \), with angles labeled and some unknown variables. The key observation is that the sum of angles around a point equals 360°.

### Given:
- \( \angle BFA = 42^\circ \)
- \( \angle BFC = 16x^\circ \)
- \( \angle DFE = (5y - 7)^\circ \)
- The right angle \( \angle AFE = 90^\circ \).

### Step-by-Step Solution:

1. **Sum of angles around point F**:  
   Since the angles form a full circle around point \( F \), they must add up to 360°. Therefore, we can set up the following equation:
   \[
   42^\circ + 16x + (5y - 7)^\circ + 90^\circ = 360^\circ
   \]

2. **Simplify the equation**:
   \[
   42 + 16x + 5y - 7 + 90 = 360
   \]
   Combine the constant terms:
   \[
   125 + 16x + 5y = 360
   \]
   Subtract 125 from both sides:
   \[
   16x + 5y = 235
   \]
   This is the first equation.

3. **Find a second equation**:  
   Notice that \( \angle DFE \) is part of the straight line \( \overrightarrow{ED} \), which means that \( \angle DFE + \angle AFE = 180^\circ \). Therefore:
   \[
   (5y - 7)^\circ + 90^\circ = 180^\circ
   \]
   Simplifying this:
   \[
   5y - 7 + 90 = 180
   \]
   \[
   5y + 83 = 180
   \]
   Subtract 83 from both sides:
   \[
   5y = 97
   \]
   Now, divide by 5:
   \[
   y = 19.4
   \]

4. **Substitute \( y = 19.4 \) into the first equation**:
   \[
   16x + 5(19.4) = 235
   \]
   Simplifying:
   \[
   16x + 97 = 235
   \]
   Subtract 97 from both sides:
   \[
   16x = 138
   \]
   Now, divide by 16:
   \[
   x = 8.625
   \]

### Final Answers:
- \( x = 8.625 \)
- \( y = 19.4 \)

Would you like more details on any step? Here are 5 related questions:

1. How do we handle equations involving angles around a point?
2. Can you explain why \( \angle DFE + \angle AFE = 180^\circ \)?
3. What is the process for solving simultaneous equations?
4. How can you check if the values of \( x \) and \( y \) are correct?
5. What role does a right angle play in geometric problems?

**Tip**: Always verify that the sum of angles around a point adds to 360° when solving such problems.

Find the values of x and y in the diagram where angles are labeled around point F, and the sum of angles around a point equals 360 degrees.

Solve for the variables x and y using the sum of angles around a point in a geometric diagram. Learn how to apply angle properties and algebra to find the solution step-by-step.

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