Math Problem Statement

Point Q is on line segment PR. Given PR = 2x, PQ = 5x - 4, and QR = x, determine the numerical length of PR

Solution

We are given that:

  • PR=2xPR = 2x
  • PQ=5x4PQ = 5x - 4
  • QR=xQR = x

Since PQ+QR=PRPQ + QR = PR (because QQ is on line segment PRPR), we can write the equation:

PQ+QR=PRPQ + QR = PR

Substituting the given values:

(5x4)+x=2x(5x - 4) + x = 2x

Now, simplify the equation:

6x4=2x6x - 4 = 2x

Subtract 2x2x from both sides:

4x4=04x - 4 = 0

Add 4 to both sides:

4x=44x = 4

Divide by 4:

x=1x = 1

Now, substitute x=1x = 1 into the expression for PRPR:

PR=2x=2(1)=2PR = 2x = 2(1) = 2

Thus, the numerical length of PRPR is 22.

Would you like more details on any step or have any further questions? Here are some related questions to explore:

  1. What would change if QRQR were given as 2x2x instead of xx?
  2. How would the solution differ if PQPQ were negative?
  3. What happens if point QQ divides PRPR in a 3:2 ratio?
  4. Can you find the lengths of PQPQ and QRQR after solving for xx?
  5. How would this process change if PRPR was an unknown constant instead of 2x2x?

Tip: When solving problems involving segments, always use the sum of smaller segments to express the full segment length as an equation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Segment Addition

Formulas

PQ + QR = PR

Theorems

Segment Addition Postulate

Suitable Grade Level

Grades 6-8