https://file.aiquickdraw.com/mathbot/file/17267080227489ary3n41.jpg

The problem in the image states:

1. Angle ABE and angle DBC are vertical angles.
2. Angle ABE and angle EBC form a linear pair.

The given equations for the measures of the angles are:
- \( m \angle ABE = 7(2x - y) - 51 \)
- \( m \angle DBC = 3x + 4(x - y) \)
- \( m \angle EBC = 4(2x + 3y) - 2(3y - x) + 1.4x - 132 \)

### Step 1: Using properties of angles
- **Vertical angles**: \( m \angle ABE = m \angle DBC \).
- **Linear pair**: \( m \angle ABE + m \angle EBC = 180^\circ \).

Let’s solve this system of equations step-by-step.

1. **Vertical angles**:
   \[
   7(2x - y) - 51 = 3x + 4(x - y)
   \]
   
2. **Linear pair**:
   \[
   [7(2x - y) - 51] + [4(2x + 3y) - 2(3y - x) + 1.4x - 132] = 180
   \]

I will now proceed to solve these equations.The solution to the system of equations is:

- \( x = 15 \)
- \( y = 18 \)

This satisfies both the conditions of vertical angles and the linear pair.

Would you like further details on the steps? Here are some related questions to expand on this:

1. How do you solve systems of equations using substitution?
2. What are vertical angles and how are they always equal?
3. Why do angles in a linear pair add up to \(180^\circ\)?
4. Can this method be applied to non-linear equations?
5. What are the properties of supplementary angles?

**Tip**: When solving systems of equations, make sure each equation is simplified as much as possible before substitution or elimination!

Angle ABE and angle DBC are vertical angles and angle ABE and angle EBC form a linear pair. m∠ABE = 7(2x − y) − 51, m∠DBC = 3x + 4(x − y), m∠EBC = 4(2x + 3y) − 2(3y − x) + 1.4x − 132. Solve for x and y.

Learn how to solve for x and y using systems of equations where angles ABE and DBC are vertical, and angles ABE and EBC form a linear pair. The problem involves applying the Vertical Angles Theorem and the Linear Pair Theorem. Suitable for high school students (Grades 9-12).

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