Math Problem Statement

ST=4x−4, T, U, equals, 4, x, minus, 10, commaTU=4x−10, and S, U, equals, 5, x, plus, 1, commaSU=5x+1, determine the numerical length of start overline, T, U, end overline, . TU

Solution

To determine the numerical length of TU\overline{TU}, given that TU=4x10TU = 4x - 10, we need to first find the value of xx.

However, from the information provided, it seems there isn't an equation directly allowing us to solve for xx unless we use some additional geometric relationships. Assuming you are working with a geometric figure like a triangle or a line segment where SS, TT, and UU are collinear points, and knowing the relationships between these points would help.

Given the information:

  • ST=4x4ST = 4x - 4
  • TU=4x10TU = 4x - 10
  • SU=5x+1SU = 5x + 1

If SS, TT, and UU are collinear and TT is between SS and UU, then SU=ST+TUSU = ST + TU.

Thus, we can set up the equation:

SU=ST+TUSU = ST + TU

Substitute the given expressions:

5x+1=(4x4)+(4x10)5x + 1 = (4x - 4) + (4x - 10)

Simplify the equation:

5x+1=8x145x + 1 = 8x - 14

Now, solve for xx:

5x8x=1415x - 8x = -14 - 1

3x=15-3x = -15

x=5x = 5

Now that we have the value of xx, substitute it back into the expression for TUTU:

TU=4x10=4(5)10=2010=10TU = 4x - 10 = 4(5) - 10 = 20 - 10 = 10

Thus, the numerical length of TU\overline{TU} is 10.

Would you like a detailed breakdown of any of these steps?

Here are some related questions:

  1. What are collinear points, and how do they influence the equation setup?
  2. How do you determine which segment is longer using algebra?
  3. What are some other geometric interpretations of SS, TT, and UU?
  4. How would the solution change if TT was not between SS and UU?
  5. How can this method be applied to other line segment problems?

Tip: Always double-check that the points are collinear or follow the specific geometric relationship before setting up equations in geometry problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Collinear Points
Segment Addition

Formulas

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Theorems

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Suitable Grade Level

High School