# If B is between A and C, AB = 3x - 1, BC = 2x + 4 \text{ and } AC = 38, what is the value of BC?

## Question:

If B is between A and C,{eq}AB = 3x - 1, BC = 2x + 4 \text{ and } AC = 38, {/eq} what is the value of BC?

## Line Segment:

A line segment has two endpoints and a fixed length. If {eq}B {/eq} is on the line segment {eq}AC {/eq} (i.e., {eq}B {/eq} lies between {eq}A {/eq} and {eq}C {/eq}), then {eq}AB+BC= AC {/eq}.

To solve the given problem, we substitute all the given values in the above equation and solve for the variable.

The given lengths are:

\begin{align} AB &= 3x - 1\\[0.3cm] BC& = 2x + 4 \\[0.3cm] AC &= 38 \end{align}

Since {eq}B{/eq} lies between {eq}A{/eq} and {eq}C{/eq}, we have:

\begin{align} AC &= AB+BC \\[0.3cm] 38 &= (3x-1)+(2x+4)\\[0.3cm] 38 &= 5x+3 \\[0.3cm] 35 &=5x& \left[ \text{ Subtracting 3 from both sides }\right] \\[0.3cm] 7&=x & \left[ \text{ Dividing both sides by 5 }\right] \\[0.3cm] \end{align}

Substitute this in the equation of {eq}BC{/eq}:

\begin{align} BC &= 2(7)+4 \\[0.3cm] &= 14+4 \\[0.3cm] &= \color{blue}{\boxed{\mathbf{18}}} \end{align}