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.