https://file.aiquickdraw.com/mathbot/file/17273730724365e8xdj5u.jpg

The problem involves a figure with two intersecting lines labeled as \( a \) and \( b \). The angle expressions are given at the intersection:

- Angle \( (5x - 7)^\circ \)
- Angle \( (3x + 17)^\circ \)
- Angle \( (4y + 3)^\circ \)

### Analysis
These angles are formed by intersecting lines, making the opposite (vertical) angles equal. Using this property, we can set up equations:

1. **Vertical Angles:**
   \[
   5x - 7 = 4y + 3
   \]

2. **Adjacent Angles (on a straight line sum to \( 180^\circ \)):**
   \[
   (5x - 7) + (3x + 17) = 180
   \]

### Solving the Equations

1. **From the adjacent angles equation:**
   \[
   (5x - 7) + (3x + 17) = 180
   \]
   Simplifying:
   \[
   8x + 10 = 180 \implies 8x = 170 \implies x = 21.25
   \]

2. **Substitute \( x = 21.25 \) into the vertical angles equation:**
   \[
   5(21.25) - 7 = 4y + 3
   \]
   Simplifying:
   \[
   106.25 - 7 = 4y + 3 \implies 99.25 = 4y + 3 \implies 4y = 96.25 \implies y = 24.06
   \]

### Conclusion
The values of \( x \) and \( y \) do not exactly match any of the options provided. However, the closest match is:

- \( x = 22.85, y = 22.86 \).

If you have any questions or need further clarification, let me know!

#### Relative Questions:
1. How do vertical angles relate to each other in intersecting lines?
2. How can we verify that the sum of angles on a straight line equals \( 180^\circ \)?
3. What would be a different approach to solving for \( x \) and \( y \) without using vertical angles?
4. What are the properties of angles formed by parallel and transversal lines?
5. How would the problem change if there were no equal angles?

#### Tip:
Always remember to check for angle relationships like vertical angles, corresponding angles, and linear pairs when dealing with intersecting or parallel lines.

Find the value of the variables in the figure where intersecting lines form angles: (5x - 7)°, (4y + 3)°, and (3x + 17)°.

Learn how to solve for x and y in an intersecting line angle problem where the angles are given by (5x - 7)°, (4y + 3)°, and (3x + 17)°. Explore key algebraic and geometric concepts like vertical angles and the sum of angles on a straight line. Suitable for students in Grades 8-10.

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