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The problem you uploaded states:

- \( BD = 7x - 10 \)
- \( BC = 4x - 29 \)
- \( CD = 5x - 9 \)

We are asked to find each value of \( BD \), \( BC \), and \( CD \).

Since \( BD \) is the entire length and includes both \( BC \) and \( CD \) (i.e., \( BD = BC + CD \)), we can set up the following equation:

\[
BD = BC + CD
\]

Substitute the given expressions:

\[
7x - 10 = (4x - 29) + (5x - 9)
\]

Now let's solve this equation step by step.

1. Combine like terms on the right-hand side:
   \[
   7x - 10 = 9x - 38
   \]

2. Subtract \( 9x \) from both sides to isolate the \( x \) terms:
   \[
   7x - 9x - 10 = -38
   \]
   \[
   -2x - 10 = -38
   \]

3. Add 10 to both sides:
   \[
   -2x = -28
   \]

4. Divide both sides by \( -2 \):
   \[
   x = 14
   \]

Now that we have the value of \( x \), we can substitute it back into the original expressions for \( BD \), \( BC \), and \( CD \) to find their values:

- \( BD = 7x - 10 = 7(14) - 10 = 98 - 10 = 88 \)
- \( BC = 4x - 29 = 4(14) - 29 = 56 - 29 = 27 \)
- \( CD = 5x - 9 = 5(14) - 9 = 70 - 9 = 61 \)

Thus, the values are:
- \( BD = 88 \)
- \( BC = 27 \)
- \( CD = 61 \)

Would you like further details or clarifications on this solution?

Here are 5 related questions for you to explore:
1. How would you solve the problem if the equation involved a quadratic expression?
2. What if the problem involved angles instead of lengths—how would the process change?
3. Can you solve for \( x \) if the equation were \( BD = BC - CD \)?
4. What geometric properties do you think can be derived from the given line segment?
5. How would you approach this problem if the values involved decimals or fractions?

**Tip:** Always double-check your work by substituting the value of \( x \) back into the original equations to verify that both sides match.

If BD = 7x - 10, BC = 4x - 29, and CD = 5x - 9, find each value.

Find the values of BD, BC, and CD when BD = 7x - 10, BC = 4x - 29, and CD = 5x - 9. This problem involves algebraic equations and the segment addition postulate. Suitable for students in Grades 7-9.

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