Math Problem Statement

If EF = 2x - 12, FG = 3x - 15, and EG = 23, find the values of x, EF, and FG.

Solution

Let's break down the problem step by step.

We are given:

EF=2x12,FG=3x15,andEG=EF+FG=23EF = 2x - 12, \quad FG = 3x - 15, \quad \text{and} \quad EG = EF + FG = 23

The total segment EGEG is the sum of EFEF and FGFG, so we can set up the following equation:

(2x12)+(3x15)=23(2x - 12) + (3x - 15) = 23

Now, simplify the left side:

2x12+3x15=232x - 12 + 3x - 15 = 23

Combine like terms:

5x27=235x - 27 = 23

Add 27 to both sides:

5x=505x = 50

Now, divide by 5:

x=10x = 10

Now that we know x=10x = 10, substitute this value into the expressions for EFEF and FGFG:

EF=2(10)12=2012=8EF = 2(10) - 12 = 20 - 12 = 8 FG=3(10)15=3015=15FG = 3(10) - 15 = 30 - 15 = 15

Thus, the values are:

  • x=10x = 10
  • EF=8EF = 8
  • FG=15FG = 15

The correct answer is: x=10,EF=8,FG=15x = 10, \quad EF = 8, \quad FG = 15

This corresponds to the first option: x=10,EF=8,FG=15x = 10, EF = 8, FG = 15.

Would you like further details or clarification? Here are five related questions to expand this concept:

  1. How do you calculate the length of a segment when multiple subsegments are given algebraically?
  2. What happens if the sum of EF+FGEF + FG is less than the total EGEG?
  3. How would the solution change if the equations for EFEF and FGFG involved different coefficients for xx?
  4. How can you visualize this problem on a number line or with a geometric drawing?
  5. What strategies can be used to check if the solution to the system of equations is correct?

Tip: Always ensure that the total segment length matches the sum of the individual parts when solving segment problems in geometry!

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Linear Equations

Formulas

EF = 2x - 12
FG = 3x - 15
EG = EF + FG

Theorems

Addition of segments in geometry

Suitable Grade Level

Grade 8-10