Question

e is the midpoint of $overline{df}$. if $de = 5x + 4$ and $df = 13x + 2$, what is $de?
simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use mid - point property

Since $E$ is the mid - point of $\overline{DF}$, we know that $DE=\frac{1}{2}DF$. So, $2DE = DF$.
Substitute $DE = 5x + 4$ and $DF=13x + 2$ into the equation: $2(5x + 4)=13x + 2$.

Step2: Expand the left - hand side

Expand $2(5x + 4)$ using the distributive property $a(b + c)=ab+ac$. We get $10x+8 = 13x + 2$.

Step3: Solve for $x$

Subtract $10x$ from both sides: $10x+8-10x=13x + 2-10x$, which simplifies to $8 = 3x+2$.
Then subtract 2 from both sides: $8 - 2=3x+2 - 2$, so $6 = 3x$.
Divide both sides by 3: $\frac{6}{3}=\frac{3x}{3}$, and $x = 2$.

Step4: Find the value of $DE$

Substitute $x = 2$ into the expression for $DE$: $DE=5x + 4$.
$DE=5\times2+4=10 + 4=14$.

Answer:

14