Problem 91 - Elementary Algebra Within Reach

Question

The remaining area of a square with side length \(x\) after a smaller square with side length \(y\) has been removed (see figure) is \((x+y)(x-y)\). (a) Show that the remaining area can also be expressed as \(x(x-y)+y(x-y)\), and give a geometric explanation for the area represented by each term in this expression. (b) Find the remaining area of a square with side length 9 after a square with side length 5 has been removed.

Step-by-Step Solution

Answer

The remaining area of a square with side length 9 after a square with side length 5 has been removed is 56 square units.

Step 1: Proof of equivalence

Expanding the second expression \(x(x-y)+y(x-y)\) using distributive property yields \(x^2-xy+xy-y^2\). Further simplifying, we have \(x^2-y^2\), which clearly matches the expression \((x+y)(x-y)\). So, the two expressions are equivalent.

Step 2: Geometric explanation

The term \(x(x-y)\) represents the area of a rectangle with sides \(x\) and \((x-y)\), which can be found by subtracting the smaller square's side length (\(y)\) from the larger square's side length (\(x)\). This represents the area of the remaining strip on two sides of the smaller square inside the larger square. Similarly, the term \(y(x-y)\) represents the area of the remaining part on the other two sides of the smaller square inside the larger square.

Step 3: Application on specific values

Substitute \(x = 9\) and \(y = 5\) into the equation \((x+y)(x-y)\). This gives \((9+5)(9-5)=14 \cdot 4=56\). So, the remaining area of the larger square after the smaller square has been removed, is 56 square units.

Key Concepts

Algebraic Expressions

Algebraic expressions are a way of representing numbers and operations using symbols and letters. They are like a mathematical shorthand that helps us simplify and solve problems. In the context of our exercise, we are manipulating these expressions to understand the concept of area left after removing a portion of a square.
One key operation used here is the distributive property. This property allows us to expand expressions like \(x(x-y)+y(x-y)\) by distributing each term across the parentheses. Specifically, distributing gives us \(xx - xy + yx - y^2\). Simplifying that, terms \(xy\) cancel out, leading to \(x^2 - y^2\). This is the same as expanding \(x+y)(x-y)\) which shows the equivalence of the two expressions. Understanding how these symbolic expressions relate to one another helps us see why they describe the same geometric area.

Geometric Interpretation

Geometric interpretation involves understanding algebraic expressions in terms of shapes and spaces. In this case, we have two squares. We start with a bigger square of side \(x\) and remove a smaller square of side \(y\).
The algebraic expression \(x(x-y)\) shows us that we're calculating the area of a rectangle. This rectangle extends from the length of the larger square's side remaining after removing the smaller square; that's \(x-y\). It accounts for the part of the bigger square that isn’t overlapped by the smaller square.
Similarly, the other term \(y(x-y)\) describes another rectangle. This rectangle forms by the width \(y\) of the smaller square overlapping with \(x-y\). It's like carving out the inner border around the smaller square. These rectangular areas together represent the remaining space within the larger square.

Area Calculation

To calculate the remaining area after removing a part of a square, we use given measurements and our equivalent algebraic expression. We take a larger square with side 9 and subtract the inner square with side 5.
Substituting into the expression \(x(x-y) + y(x-y)\), we see it reduces to evaluating \( (9+5)(9-5) \). This simplifies to \( 14 \times 4 = 56 \). This is the amount of space that remains outside of the inner square but still within the outer square.
Area calculations are important in real-world applications too, such as finding usable space in buildings or materials. The neat alignment of algebra with geometry allows these theoretical ideas to translate into practical results, letting us understand and calculate space efficiently.