Question

find the value of x so that the area of the rectangle is equal to the area of the triangle.

Explanation:

Step1: Calculate rectangle area

The area formula for a rectangle is $A = l\times w$. Here, $l=x + 5$ and $w = 6$, so $A_{rectangle}=6(x + 5)=6x+30$.

Step2: Calculate triangle area

The area formula for a triangle is $A=\frac{1}{2}\times b\times h$. Here, $b = 12$ and $h=2x - 2$, so $A_{triangle}=\frac{1}{2}\times12\times(2x - 2)=6(2x - 2)=12x-12$.

Step3: Set areas equal and solve for x

Set $A_{rectangle}=A_{triangle}$, so $6x + 30=12x-12$.
Subtract $6x$ from both sides: $30 = 6x-12$.
Add 12 to both sides: $42=6x$.
Divide both sides by 6: $x = 7$.

Answer:

$7$