The image shows two intersecting lines with angles labeled at their intersections. The angles $(4x + 7)^\circ$ and $(10x + 5)^\circ$ are corresponding angles. Because these lines appear to be parallel and cut by a transversal, corresponding angles are equal.

Thus, we can set up the equation: $(4x + 7)^\circ = (10x + 5)^\circ$

Let's solve for $x$:

1. Subtract $4x$ from both sides: $7 = 6x + 5$

2. Subtract 5 from both sides: $2 = 6x$

3. Divide by 6: $x = \frac{2}{6} = \frac{1}{3}$

So, $x = \frac{1}{3}$.

Would you like more details or explanations on this? Here are some related questions:

1. What are corresponding angles, and why are they equal for parallel lines?
2. How do we know the lines are parallel in this image?
3. Can you provide an example where corresponding angles are not equal?
4. What happens if the angles were supplementary instead of equal?
5. How can we verify the solution for $x$ using substitution?

Tip: In problems involving parallel lines and transversals, always look for angle pairs like corresponding, alternate interior, or same-side interior angles to form your equations.