Math Problem Statement

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 .

Solution

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

  1. CD=2xCD = 2x
  2. BD=3x+4BD = 3x + 4
  3. BC=2x1BC = 2x - 1

Since point CC is on the line segment BD\overline{BD}, the sum of the segments BCBC and CDCD should equal the total length of BDBD. Therefore, we can write the equation:

BC+CD=BDBC + CD = BD

Substituting the given expressions for BCBC, CDCD, and BDBD:

(2x1)+2x=3x+4(2x - 1) + 2x = 3x + 4

Now, simplify the left-hand side:

2x1+2x=4x12x - 1 + 2x = 4x - 1

Thus, the equation becomes:

4x1=3x+44x - 1 = 3x + 4

Now, solve for xx:

4x3x=4+14x - 3x = 4 + 1

x=5x = 5

Now that we have the value of xx, we can substitute it into the expression for BDBD:

BD=3x+4=3(5)+4=15+4=19BD = 3x + 4 = 3(5) + 4 = 15 + 4 = 19

Thus, the length of BD\overline{BD} is 19\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+2BD = 4x + 2?
  2. What is the length of BCBC when x=5x = 5?
  3. How can we verify that BC+CD=BDBC + CD = BD holds for any xx?
  4. What happens if point CC is the midpoint of BD\overline{BD}?
  5. Can you find the value of CDCD when x=5x = 5?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Geometry
Segments on a Line

Formulas

BC + CD = BD
Solving linear equations

Theorems

Segment Addition Postulate

Suitable Grade Level

Grades 6-8