Solving Geometry Problems with Algebra: Finding the Remaining Area of a Rectangle

In geometry, problems involving rectangles and their areas can often be solved using algebraic expressions. In this tutorial, we will walk through how to determine the remaining area of a rectangle after removing a smaller rectangle from the original. We will follow a step-by-step process to solve a specific problem and clarify the common mistakes in such calculations.

The Original Problem

The original rectangle has a length of 7x – 5 and a width of 5x – 2. Cameron cuts out a smaller rectangle with a side length of 2x 3. What expression represents the remaining area of the original rectangle?

Step 1: Calculate the Area of the Original Rectangle

The area A of a rectangle is given by the formula:

A length times; width

For the original rectangle:

So the area of the original rectangle is:

Aoriginal (7x – 5) times; (5x – 2)

Expanding this expression:

Aoriginal 7x times; 5x – 7x times; 2 – 5 times; 5x 5 times; 2

Aoriginal 35x2 – 14x – 25x 10

Aoriginal 35x2 – 39x 10

Step 2: Calculate the Area of the Cut-out Rectangle

The cut-out rectangle has a side length of 2x 3. Assuming it is a square (since the problem does not specify otherwise), its area is:

Acut (2x 3) times; (2x 3)

Acut 4x2 6x 6x 9

Acut 4x2 12x 9

Step 3: Calculate the Remaining Area

Subtract the area of the cut-out rectangle from the area of the original rectangle:

Aremaining Aoriginal - Acut

Aremaining (35x2 – 39x 10) - (4x2 12x 9)

Aremaining 35x2 - 4x2 – 39x - 12x 10 - 9

Aremaining 31x2 - 51x 1

Therefore, the expression representing the remaining area of the original rectangle after Cameron cuts out the smaller rectangle is:

boxed{31x^2 - 51x 1}

A Common Mistake

The mistake in the provided solution comes from cutting off 2x 3 from the length or the width instead of the width since it was specified to cut out 2x 3 from the side (which could be either length or width).

Correct Solution

To correctly solve the problem, we need to consider that the cut is made from the width:

New Area 7x times; 3x – 7x times; 5 – 5 times; 3x 5 times; 5

New Area 21x2 - 35x – 15x 25

New Area 21x2 - 5 25

Hence, the correct remaining area expression is:

boxed{21x^2 - 5 25}