https://file.aiquickdraw.com/mathbot/file/1725580070471h7jpqmhc.jpg

The problem in the image shows a rectangular shape with segment \( DF \), where you are asked to find the length of segment \( EF \).

The information provided is:
- \( DF = 9x - 39 \)
- \( EF = 3x + 10 \)
- \( D \) is located at 47 on the number line.

I will now solve for the value of \( EF \) based on this information. 

The complete distance \( DF \) can be written as the sum of two parts, namely the segment \( DE \) and \( EF \). From the image, it appears \( DF = 47 \), and we have the equations for both segments involving \( x \).

So, we can set up the equation:
\[
DF = DE + EF
\]

Next, let me calculate \( EF \).The length of segment \( EF \) is \( \frac{116}{3} \), or approximately 38.67 units.

Would you like more details or have any questions about this? Here are some related questions to consider:

1. How do you calculate the length of \( DF \) given different values of \( x \)?
2. What happens to \( EF \) if the equation for \( DF \) changes?
3. Can you solve for the value of \( x \) in terms of \( DF \) if the total length \( DF \) is unknown?
4. How would you approach this problem if the segments were positioned differently in the rectangle?
5. What is the significance of solving such linear equations in geometry?

**Tip**: In problems involving variables and equations, it's helpful to set up clear relationships between known and unknown quantities first, and then solve step-by-step.

Find the length of segment EF given DF = 9x - 39, EF = 3x + 10, and D is located at 47 on the number line.

This problem involves finding the length of segment EF based on the given equations DF = 9x - 39 and EF = 3x + 10. Step-by-step instructions are provided, along with concepts of linear equations and geometry. Suitable for students in Grades 8-10.

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