Question

e is the midpoint of \\(\overline{df}\\). if \\(de = x + 7\\) and \\(df = 5x - 1\\), what is \\(df\\)? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( E \) is the midpoint of \( \overline{DF} \), we know that \( DE = \frac{1}{2}DF \). So we can set up the equation: \( x + 7=\frac{1}{2}(5x - 1) \).

Step2: Solve the equation for \( x \)

Multiply both sides of the equation by 2 to get rid of the fraction: \( 2(x + 7)=5x - 1 \).
Expand the left - hand side: \( 2x+14 = 5x-1 \).
Subtract \( 2x \) from both sides: \( 14=5x - 2x-1 \), which simplifies to \( 14 = 3x-1 \).
Add 1 to both sides: \( 14 + 1=3x \), so \( 15 = 3x \).
Divide both sides by 3: \( x=\frac{15}{3}=5 \).

Step3: Find the value of \( DF \)

We know that \( DF = 5x-1 \). Substitute \( x = 5 \) into the expression for \( DF \): \( DF=5\times5 - 1 \).
Calculate \( 5\times5=25 \), then \( 25-1 = 24 \).

Answer:

24