Q22. - Algebra 1

Question

GEOMETRY Find the value of x so the rectangles have the same area.

Step-by-Step Solution

Answer

The value of x so that the figures have the same area is 8.

Step 1. State the formula.

The area of a rectangle is given as $lw$, where $l$ is the length of the rectangle and  is the width of the rectangle.

Step 2. List the given data.

From the given figure, the length of the first rectangle is $x$ units and the width of the rectangle is $12$ units. Then, the area of the first rectangle is $12x$ units.

 

Similarly, from the given figure, the length of the second rectangle is $16$ units and the width of the second rectangle is $x-2$ units. Then, the area of the second rectangle is $16\left(x-2\right)$ units.

Step 3. Formulate the equation.

It is given that the figures have the same area. Then, the areas of the first and the second rectangle are equal. So, “$12x=16\left(x-2\right)$”.

 

This is the required equation.

Step 4. Solve the obtained equation.

Solving,

$12x=16\left(x-2\right)$  (Given equation)

$12x=16x-32$  (Distributive property)

$12x-16x=16x-32-16x$  (Subtract $16x$ from both sides)

$-4x=-32$  (Simplify)

$\frac{-4x}{-4}=\frac{-32}{-4}$  (Divide both sides by $-4$)

$x=8$  (Simplify)

So, $x=8$ is the solution of the given equation.

Step 5. Interpret the obtained solution.

It was assumed that the areas of both figures are equal. This assumption led to the obtained solution $x=8$.

 

Thus, the value of x so that the figures have the same area is 8.