Question

d is the midpoint of $overline{ce}$. if $de = 10x - 5$ and $ce = 19x - 5$, what is $de? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use mid - point property

Since \(D\) is the mid - point of \(\overline{CE}\), then \(CE = 2DE\).
We know that \(DE=10x - 5\) and \(CE = 19x - 5\), so \(19x-5=2(10x - 5)\).

Step2: Expand the right - hand side

Expand \(2(10x - 5)\) using the distributive property \(a(b - c)=ab-ac\). Here \(a = 2\), \(b = 10x\), \(c = 5\), so \(2(10x - 5)=20x-10\).
The equation becomes \(19x-5=20x - 10\).

Step3: Solve for \(x\)

Subtract \(19x\) from both sides of the equation:
\(19x-5-19x=20x - 10-19x\).
\(-5=x - 10\).
Then add 10 to both sides: \(-5 + 10=x-10 + 10\), so \(x = 5\).

Step4: Find the value of \(DE\)

Substitute \(x = 5\) into the expression for \(DE\).
\(DE=10x-5=10\times5-5\).
\(DE = 50 - 5=45\).

Answer:

45