To solve for the length of [tex]\(\overline{UV}\)[/tex], we need to use the given expressions for the segments and the fact that [tex]\(V\)[/tex] is somewhere on [tex]\(\overline{UW}\)[/tex]. Here are the steps:
1. Express the Segments:
- [tex]\(UV = 2x - 13\)[/tex]
- [tex]\(VW = -18 + 2x\)[/tex]
- [tex]\(UW = 17\)[/tex]
2. Set Up the Equation:
Since [tex]\(V\)[/tex] lies on [tex]\(\overline{UW}\)[/tex], the sum of the segments [tex]\(UV\)[/tex] and [tex]\(VW\)[/tex] must equal the total length of [tex]\(\overline{UW}\)[/tex].
[tex]\[
UV + VW = UW
\][/tex]
Substitute the given expressions into the equation:
[tex]\[
(2x - 13) + (-18 + 2x) = 17
\][/tex]
3. Simplify the Equation:
Combine like terms:
[tex]\[
2x - 13 - 18 + 2x = 17
\][/tex]
[tex]\[
4x - 31 = 17
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Isolate [tex]\(x\)[/tex] by adding 31 to both sides of the equation:
[tex]\[
4x = 48
\][/tex]
Then divide both sides by 4:
[tex]\[
x = 12
\][/tex]
5. Calculate the Length of [tex]\(UV\)[/tex]:
Substitute [tex]\(x = 12\)[/tex] back into the expression for [tex]\(UV\)[/tex]:
[tex]\[
UV = 2x - 13
\][/tex]
[tex]\[
UV = 2(12) - 13
\][/tex]
[tex]\[
UV = 24 - 13
\][/tex]
[tex]\[
UV = 11
\][/tex]
Therefore, the length of [tex]\(\overline{UV}\)[/tex] is [tex]\(11\)[/tex]. The correct answer is [tex]\(11\)[/tex].
