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!