https://file.aiquickdraw.com/mathbot/file/1726483596129cs8lmeac.jpg

In this image, the problem involves two expressions representing angles formed by intersecting lines. These angles are vertically opposite, which means they are equal. The two expressions are:

- \(18x + 12\)
- \(24x - 13\)

Since vertically opposite angles are equal, we can set up the equation:

\[
18x + 12 = 24x - 13
\]

### Step-by-step solution:

1. **Set up the equation:**

\[
18x + 12 = 24x - 13
\]

2. **Move all the terms involving \(x\) to one side:**
   
\[
18x - 24x = -13 - 12
\]

\[
-6x = -25
\]

3. **Solve for \(x\):**

\[
x = \frac{-25}{-6} = \frac{25}{6} \approx 4.17
\]

So, \(x \approx 4.17\).

### Calculating angle values:

- For the first angle: \( 18x + 12 = 18(4.17) + 12 \approx 87.06 + 12 = 99.06^\circ \)
- For the second angle: \( 24x - 13 = 24(4.17) - 13 \approx 100.08 - 13 = 99.08^\circ \)

Both angles are approximately \(99^\circ\), confirming that they are equal.

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. What is the concept of vertically opposite angles?
2. How do we handle equations with variables on both sides?
3. Why are vertically opposite angles always equal?
4. How do you verify your solution after finding \(x\)?
5. What are some applications of this concept in real life?

**Tip:** When solving equations with variables on both sides, always move the terms involving the variable to one side and constants to the other side to isolate the variable easily.

Solve for x given that 18x + 12 and 24x - 13 represent vertically opposite angles.

Learn how to solve for x when given two expressions, 18x + 12 and 24x - 13, representing vertically opposite angles. This problem involves algebraic manipulation and the use of the vertically opposite angles theorem, suitable for students in Grades 7-10.

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