Let's revisit the problem with the correct expressions:

C is a point on segment $\overline{BD}$.

• $BC = 20x$
• $CD = 9$
• $BD = 13x + 16$

We need to find $BC$.

Step 1: Use the segment addition postulate.

Since $C$ is a point on segment $\overline{BD}$, the entire segment $BD$ is the sum of $BC$ and $CD$: $BD = BC + CD$

Substituting the given values: $13x + 16 = 20x + 9$

Step 2: Solve for $x$.

First, subtract $20x$ from both sides: $13x + 16 - 20x = 9$ $-7x + 16 = 9$

Next, subtract 16 from both sides: $-7x = 9 - 16$ $-7x = -7$

Now, divide by $-7$: $x = 1$

Step 3: Find $BC$.

Since $BC = 20x$, substitute $x = 1$ into the equation: $BC = 20 \times 1 = 20$

So, the length of $BC$ is $\boxed{20}$.

Would you like further details or have any questions?

Here are five related questions you might consider:

1. What if $CD$ were given as a function of $x$ instead of a constant?
2. How would the solution change if $BD$ were a more complex expression, such as a quadratic function of $x$?
3. What if the problem included an additional segment on $BD$? How would that affect the equation?
4. Can you confirm the solution by checking if the calculated $BC$ and $CD$ add up to $BD$?
5. What strategies can you use to simplify similar equations involving segments?

Tip: Always make sure to carefully substitute and solve for variables step by step, as small arithmetic errors can lead to incorrect solutions.