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Given that lines \( u \) and \( v \) are parallel, the corresponding angles formed by the transversal are equal. In this case, the angles \( 6x + 7 \) and \( 5x + 15 \) are corresponding angles. 

### Setting the corresponding angles equal:
\[
6x + 7 = 5x + 15
\]

### Solving for \( x \):
1. Subtract \( 5x \) from both sides:
\[
6x - 5x + 7 = 15
\]
\[
x + 7 = 15
\]

2. Subtract 7 from both sides:
\[
x = 8
\]

Thus, the value of \( x \) is 8.

Would you like any further details or have any questions?

Here are some related questions:

1. What are corresponding angles, and why are they equal for parallel lines?
2. How can you prove that two lines are parallel using angles?
3. What is the role of a transversal line in geometry?
4. How do alternate interior angles behave when two lines are parallel?
5. Can the value of \( x \) be different if the lines were not parallel?

**Tip:** Always check if angles are corresponding, alternate interior, or supplementary when dealing with parallel lines and a transversal.

Learn how to solve for x when two lines u and v are parallel and intersected by a transversal. This problem involves corresponding angles, parallel line properties, and solving linear equations. Suitable for students in Grades 8-10.

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