In the diagram, we see that there is an angle at point $C$. The angles $\angle BCD$ (a straight angle) and $\angle ECD$ (labeled as $10x + 5^\circ$) add up to a straight line, which means their sum must be $180^\circ$.

The angle $\angle BCD = 75^\circ$ (as given), and the other angle $\angle ECD = 10x + 5^\circ$.

Using the fact that these angles form a straight line: $75^\circ + (10x + 5^\circ) = 180^\circ$

Simplify the equation: $75 + 10x + 5 = 180$ $10x + 80 = 180$

So the correct equation to solve for $x$ is $10x + 80 = 180$.

Now, let's solve for $x$: $10x + 80 = 180$ $10x = 180 - 80$ $10x = 100$ $x = \frac{100}{10} = 10$

Thus, $x = 10$.

Let me know if you'd like more details or explanations! Here are 5 related questions to extend this concept:

1. What are supplementary angles, and how do they relate to straight lines?
2. How would the equation change if the angle was $12x + 5^\circ$ instead?
3. What is the general method to solve for an unknown in a linear equation?
4. Can you derive $x$ if the angle was $(15x + 10)^\circ$ instead of $10x + 5^\circ$?
5. How do you verify that your solution for $x$ is correct in this type of problem?

Tip: When working with angles on a straight line, always remember their sum is $180^\circ$. This helps in setting up equations easily.