https://file.aiquickdraw.com/mathbot/file/1726595728352xx0xg70i.jpg

The question asks to find the values of \( x \) and \( y \) using the given diagram of angles, which appear to be linear pairs. Let’s analyze the information provided in the diagram:

- The angle in the first quadrant is labeled \( (2x + 25)^\circ \).
- The angle in the second quadrant is labeled \( y^\circ \).
- The angle in the third quadrant is labeled \( (3x - 10)^\circ \).

### Step-by-Step Solution:

1. **Identify Relationships between Angles:**
   Since the angles form a complete revolution around a point (a full circle), the sum of all four angles must equal \( 360^\circ \).

   The full circle consists of two pairs of linear angles (straight lines), so:
   \[
   (2x + 25)^\circ + y^\circ = 180^\circ \quad \text{(linear pair)}
   \]
   \[
   (3x - 10)^\circ + y^\circ = 180^\circ \quad \text{(linear pair)}
   \]

2. **Set up the Equations:**

   - From the first pair:
     \[
     (2x + 25) + y = 180
     \]
     Simplifying:
     \[
     2x + y = 155 \quad \text{(Equation 1)}
     \]

   - From the second pair:
     \[
     (3x - 10) + y = 180
     \]
     Simplifying:
     \[
     3x + y = 190 \quad \text{(Equation 2)}
     \]

3. **Solve the System of Equations:**

   Subtract Equation 1 from Equation 2:
   \[
   (3x + y) - (2x + y) = 190 - 155
   \]
   Simplifying:
   \[
   x = 35
   \]

4. **Substitute \( x = 35 \) into one of the original equations:**

   Using Equation 1:
   \[
   2(35) + y = 155
   \]
   Simplifying:
   \[
   70 + y = 155
   \]
   \[
   y = 85
   \]

Thus, the values are:
\[
x = 35, \quad y = 85
\]

Would you like further details on any step? 

### Related Questions:
1. How are linear pairs of angles defined?
2. What is the sum of angles in a complete revolution?
3. How can we solve systems of equations algebraically?
4. How do angle relationships change when formed by intersecting lines?
5. Can angles on a straight line always sum to 180 degrees?

**Tip:** For linear pair problems, remember that adjacent angles formed by intersecting lines always sum to 180 degrees.

Explore how to find the values of x and y using linear pair angles in a full circle. This guide provides a step-by-step solution to the equations (2x + 25) + y = 180 and (3x - 10) + y = 180, using basic algebra and geometry concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation