If BC = 7, CD = x + 5, and BD = 7x, what is CD?

Step-by-step solution

1. 1

   Write the postulate for this labelling. $C$ lies between $B$ and $D$, so

   $BC+CD=BD$

   Read the letters carefully: the segment named by the outer two points, $BD$, is the whole.

2. 2

   Substitute the three expressions.

   $7+(x+5)=7x\;\Longrightarrow\;x+12=7x$

3. 3

   Isolate $x$. Subtract $x$ from both sides and divide by the coefficient:

   $12=6x\;\Longrightarrow\;x=2$

4. 4

   Answer the question that was asked. The target is $CD$, not $x$:

   $CD=x+5=2+5=7$

5. 5

   Check the arithmetic and interpret. With $x=2$: $BC=7$, $CD=7$, $BD=7(2)=14$, and $7+7=14\;\checkmark$. Because $BC=CD$, the point $C$ is the midpoint of $\overline{BD}$ — a useful consistency signal, since two equal pieces and a whole of exactly twice their length must agree.