Point V is on line segment start overline, U, W, end overline*UW*. Given U, V, equals, 6, comma*UV*=6, U, W, equals, 5, x, comma*UW*=5*x*, and V, W, equals, 4, x, minus, 1, comma*VW*=4*x*−1, determine the numerical length of start overline, V, W, end overline, .*VW*.

We are given that point \( V \) is on line segment \( \overline{UW} \), with the following information:

- \( UV = 6 \),
- \( UW = 5x \),
- \( VW = 4x - 1 \).

Since \( V \) is on the line segment \( \overline{UW} \), we know that:

\[
UV + VW = UW.
\]

Substitute the given values into this equation:

\[
6 + (4x - 1) = 5x.
\]

Now, simplify and solve for \( x \):

\[
6 + 4x - 1 = 5x,
\]
\[
5 + 4x = 5x,
\]
\[
5 = 5x - 4x,
\]
\[
5 = x.
\]

Now that we have \( x = 5 \), substitute this back into the expression for \( VW \):

\[
VW = 4x - 1 = 4(5) - 1 = 20 - 1 = 19.
\]

Thus, the numerical length of \( \overline{VW} \) is \( \boxed{19} \).

Would you like more details or have any questions? Here are 5 related questions you could explore:

1. How would you solve for \( UV \) if \( VW \) were unknown?
2. What is the length of the entire line segment \( UW \)?
3. If \( V \) were not on the line segment \( UW \), how would that affect the solution?
4. How would the equation change if \( UV \) were a function of \( x \)?
5. Can you express the position of \( V \) on \( UW \) as a fraction of \( UW \)?

**Tip:** When solving for unknowns in geometry problems, always write down known relationships, such as segment addition postulates, to simplify the solution!

Point V is on line segment UW. Given UV = 6, UW = 5x, and VW = 4x - 1, determine the numerical length of VW.

Determine the numerical length of segment VW using algebraic expressions and the segment addition postulate. This problem involves solving for x and applying it to find the length of VW. Suitable for students in Grades 8-10.

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