Question

1. if $overline{bd} perp overline{ac}$, $mangle dbe=(2x - 1)^{circ}$, and $mangle cbe=(5x - 42)^{circ}$, find the value of $x$.

Explanation:

Step1: Identify angle - relationship

Since $\overline{BD}\perp\overline{AC}$, $\angle DBC = 90^{\circ}$. And $\angle CBE=\angle DBC+\angle DBE$. So, $(5x - 42)^{\circ}=(2x - 1)^{\circ}+90^{\circ}$.

Step2: Solve the equation

First, expand the right - hand side: $5x-42 = 2x-1 + 90$.
Then, simplify the right - hand side: $5x-42=2x + 89$.
Next, move the terms with $x$ to one side: $5x-2x=89 + 42$.
Combine like terms: $3x=131$.
Finally, solve for $x$: $x=\frac{131}{3}$.

Answer:

$x=\frac{131}{3}$