Point C is on line segment start overline, B, D, end overline 
BD
 . Given C, D, equals, 2, x, commaCD=2x, B, D, equals, 3, x, plus, 4, commaBD=3x+4, and B, C, equals, 2, x, minus, 1, commaBC=2x−1, determine the numerical length of start overline, B, D, end overline, . 
BD
 .

We are given three relationships between the line segments on line segment \( \overline{BD} \):

1. \( CD = 2x \)
2. \( BD = 3x + 4 \)
3. \( BC = 2x - 1 \)

Since point \( C \) is on the line segment \( \overline{BD} \), the sum of the segments \( BC \) and \( CD \) should equal the total length of \( BD \). Therefore, we can write the equation:

\[
BC + CD = BD
\]

Substituting the given expressions for \( BC \), \( CD \), and \( BD \):

\[
(2x - 1) + 2x = 3x + 4
\]

Now, simplify the left-hand side:

\[
2x - 1 + 2x = 4x - 1
\]

Thus, the equation becomes:

\[
4x - 1 = 3x + 4
\]

Now, solve for \( x \):

\[
4x - 3x = 4 + 1
\]

\[
x = 5
\]

Now that we have the value of \( x \), we can substitute it into the expression for \( BD \):

\[
BD = 3x + 4 = 3(5) + 4 = 15 + 4 = 19
\]

Thus, the length of \( \overline{BD} \) is \( \boxed{19} \).

Would you like more details or have any further questions? Here are five related questions for deeper understanding:

1. How would the problem change if \( BD = 4x + 2 \)?
2. What is the length of \( BC \) when \( x = 5 \)?
3. How can we verify that \( BC + CD = BD \) holds for any \( x \)?
4. What happens if point \( C \) is the midpoint of \( \overline{BD} \)?
5. Can you find the value of \( CD \) when \( x = 5 \)?

**Tip:** When solving for unknowns, ensure that all parts of the equation correspond to the correct geometric interpretation.

Point C is on line segment BD. Given CD = 2x, BD = 3x + 4, and BC = 2x - 1, determine the numerical length of BD.

Determine the length of line segment BD using algebraic expressions for BC, CD, and BD. This problem involves key algebraic concepts like solving linear equations and understanding the Segment Addition Postulate. Suitable for students in Grades 6-8.

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