If DE = 12x − 2, EF = 17x − 13, and DF = 14, what is EF?

12x − 2
D
17x − 13
E
F
14


Simplify your answer and write it as a proper fraction, mixed number, or integer.


To solve for \( EF \) given the equations \( DE = 12x - 2 \), \( EF = 17x - 13 \), and \( DF = 14 \), we can use the relationship between the segments.

Since \( DF = DE + EF \), we can set up the equation:

\[
DF = DE + EF
\]

Substituting the given values:

\[
14 = (12x - 2) + (17x - 13)
\]

Now, simplify the right side:

\[
14 = 12x - 2 + 17x - 13
\]
\[
14 = 29x - 15
\]

Next, isolate \( x \):

\[
14 + 15 = 29x
\]
\[
29 = 29x
\]
\[
x = 1
\]

Now that we have \( x \), we can find \( EF \):

\[
EF = 17x - 13
\]
Substituting \( x = 1 \):

\[
EF = 17(1) - 13 = 17 - 13 = 4
\]

Thus, the value of \( EF \) is:

\[
\boxed{4}
\]

If you have any questions or want more details, let me know! Here are some follow-up questions:

1. What is the geometric interpretation of segments \( DE \), \( EF \), and \( DF \)?
2. How would the values change if \( DE \) were doubled?
3. Can you provide an example of a similar problem?
4. What would happen if \( DF \) was increased to 20?
5. How can you apply this method to solve for unknowns in other geometric figures?

**Tip:** Always ensure to isolate variables step-by-step for clarity in solving equations.

If DE = 12x − 2, EF = 17x − 13, and DF = 14, what is EF?

Discover how to find the length of segment EF given DE = 12x - 2, EF = 17x - 13, and DF = 14. This step-by-step solution utilizes algebraic concepts and the segment addition postulate, suitable for students in Grades 7-9.

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