Math Problem Statement

Point V is on line segment start overline, U, W, end overlineUW. Given U, V, equals, 6, commaUV=6, U, W, equals, 5, x, commaUW=5x, and V, W, equals, 4, x, minus, 1, commaVW=4x−1, determine the numerical length of start overline, V, W, end overline, .VW.

Solution

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

  • UV=6UV = 6,
  • UW=5xUW = 5x,
  • VW=4x1VW = 4x - 1.

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

UV+VW=UW.UV + VW = UW.

Substitute the given values into this equation:

6+(4x1)=5x.6 + (4x - 1) = 5x.

Now, simplify and solve for xx:

6+4x1=5x,6 + 4x - 1 = 5x, 5+4x=5x,5 + 4x = 5x, 5=5x4x,5 = 5x - 4x, 5=x.5 = x.

Now that we have x=5x = 5, substitute this back into the expression for VWVW:

VW=4x1=4(5)1=201=19.VW = 4x - 1 = 4(5) - 1 = 20 - 1 = 19.

Thus, the numerical length of VW\overline{VW} is 19\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 UVUV if VWVW were unknown?
  2. What is the length of the entire line segment UWUW?
  3. If VV were not on the line segment UWUW, how would that affect the solution?
  4. How would the equation change if UVUV were a function of xx?
  5. Can you express the position of VV on UWUW as a fraction of UWUW?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Geometry
Segment Addition Postulate

Formulas

UV + VW = UW

Theorems

Segment Addition Postulate

Suitable Grade Level

Grades 8-10