https://file.aiquickdraw.com/mathbot/file/1726446655654vezfrc93.jpg

We are given that the total length \( DF = 42 \), and we need to find \( DE \). The figure shows that the segment \( DF \) is composed of two smaller segments: \( DE = 7x + 1 \) and \( EF = 4x - 3 \).

The sum of these two segments is equal to the total length:

\[
DE + EF = DF
\]

Substituting the expressions for \( DE \) and \( EF \):

\[
(7x + 1) + (4x - 3) = 42
\]

Now, let's solve this equation for \( x \).

### Step 1: Combine like terms
\[
7x + 4x + 1 - 3 = 42
\]
\[
11x - 2 = 42
\]

### Step 2: Solve for \( x \)
Add 2 to both sides:
\[
11x = 44
\]

Divide by 11:
\[
x = 4
\]

### Step 3: Find \( DE \)
Now that we know \( x = 4 \), substitute this value back into the expression for \( DE \):
\[
DE = 7x + 1 = 7(4) + 1 = 28 + 1 = 29
\]

Thus, \( DE = 29 \).

Let me know if you need more details or have any questions! Here are some related questions to consider:

1. What is the length of \( EF \)?
2. How would the problem change if \( DF \) were a different length?
3. Can we use a different method to solve for \( DE \)?
4. What happens to \( DE \) if the values of \( x \) are negative?
5. How can you check if your solution is correct using different strategies?

**Tip**: Always check if the sum of segment lengths adds up to the total given length as a verification step!

If DF = 42, find DE. The line segment DF is composed of DE = 7x + 1 and EF = 4x - 3.

In this algebra problem, learn how to solve for DE when DF = 42 using linear equations. The total length DF is broken into two parts: DE = 7x + 1 and EF = 4x - 3. Suitable for Grades 7-9.

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