The problem provides the following information:

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

To find $EF$, we need to use the fact that $DF = DE + EF$, which gives us the equation:

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

Now, we will solve for $x$.

$9x - 39 = 47 + 3x + 10$ $9x - 39 = 57 + 3x$ $9x - 3x = 57 + 39$ $6x = 96$ $x = \frac{96}{6} = 16$

Now that we have $x = 16$, we can substitute this value back into the expression for $EF$.

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

Thus, the length of $EF$ is 58 units.

Would you like any more details on the solution, or do you have any questions?

Related Questions:

1. How would the solution change if $DF = 9x + 39$?
2. How do we verify the value of $x$ by substituting it back into the equation for $DF$?
3. What if $DE$ and $EF$ were not given in terms of expressions but just values?
4. How do we solve a similar problem where all segments are expressions?
5. What is the importance of understanding segment addition in geometry?

Tip: Always make sure to double-check the steps when solving for unknowns in geometric problems to avoid minor errors!