Question

Point B is the midpoint start overline, A, C, end overline
AC
. If AB = x + 15 and BC = 5x -1, find the value of x and AC

Answer 1

To find the value of \( x \) and the length of line segment \( AC \), we can use the fact that point \( B \) is the midpoint of line segment \( AC \). This means that the lengths \( AB \) and \( BC \) are equal.

Given:
\[
AB = x + 15
\]
\[
BC = 5x - 1
\]

Since \( B \) is the midpoint:
\[
AB = BC
\]

Setting the two expressions for \( AB \) and \( BC \) equal to each other gives:
\[
x + 15 = 5x - 1
\]

Now, we can solve for \( x \). First, we can move all terms involving \( x \) to one side and constant terms to the other side:
\[
15 + 1 = 5x - x
\]
\[
16 = 4x
\]

Now, divide both sides by 4:
\[
x = 4
\]

Next, we'll find the lengths of \( AB \) and \( BC \) using \( x = 4 \):

Calculating \( AB \):
\[
AB = x + 15 = 4 + 15 = 19
\]

Calculating \( BC \):
\[
BC = 5x - 1 = 5(4) - 1 = 20 - 1 = 19
\]

Both segments \( AB \) and \( BC \) indeed equal 19.

Now, to find the length of \( AC \):
\[
AC = AB + BC = 19 + 19 = 38
\]

In conclusion:
- The value of \( x \) is \( 4 \).
- The length of \( AC \) is \( 38 \).