Math Problem Statement

If DF = 9x - 39, find EF. DF = DE + EF, where DE = 47 and EF = 3x + 10.

Solution

Let's solve the problem in the image.

We are given:

  • DF=9x39DF = 9x - 39
  • DE=47DE = 47
  • EF=3x+10EF = 3x + 10

The relationship is:

DF=DE+EFDF = DE + EF

Substituting the given values:

9x39=47+(3x+10)9x - 39 = 47 + (3x + 10)

Now, let's solve this equation:

9x39=47+3x+109x - 39 = 47 + 3x + 10

Simplifying the right side:

9x39=57+3x9x - 39 = 57 + 3x

Subtract 3x3x from both sides:

6x39=576x - 39 = 57

Add 39 to both sides:

6x=966x = 96

Divide by 6:

x=16x = 16

Now that we know x=16x = 16, let's find EFEF.

Substitute x=16x = 16 into EF=3x+10EF = 3x + 10:

EF=3(16)+10=48+10=58EF = 3(16) + 10 = 48 + 10 = 58

So, EF=58EF = 58.

Would you like any more details or have any questions?

Here are five related questions to expand on this problem:

  1. How would the result change if EFEF was expressed as 2x+202x + 20 instead of 3x+103x + 10?
  2. What would happen if DFDF was expressed in a quadratic form?
  3. Can you solve for EFEF if DFDF were given as a constant number, say 150?
  4. What if the equation included more segments on the line, such as an additional point between DD and EE?
  5. How do equations change if distances between points are defined in terms of different variables?

Tip: Always combine like terms and simplify the equation step by step to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

DF = DE + EF

Theorems

Equation Balancing Theorem

Suitable Grade Level

Grades 8-10