Problem 91 - Elementary Algebra Within Reach

Question

Evaluate the algebraic expression for the specified values of the variables. \(x^{2}-y^{2}\) (a) \(x=4, y=3\) (b) \(x=-5, y=3\)

Step-by-Step Solution

Answer

The answer for (a) \(x=4, y=3\) is 7, and the answer for (b) \(x=-5, y=3\) is 16.

Step 1: Substitution for Case (a)

Firstly, substitute the given values into the algebraic expression. For \(x=4, y=3\) in \(x^{2}-y^{2}\), it becomes \(4^{2} - 3^{2}\).

Step 2: Evaluation for Case (a)

Following PEMDAS/BODMAS, squares come first. Calculate 4 squared (16) and 3 squared (9) separately. Then subtract these values: \(16 - 9 = 7\). So, for \(x=4, y=3\), the expression \(x^{2} - y^{2}\) evaluates to 7.

Step 3: Substitution for Case (b)

Next, substitute the given values into the algebraic expression for the second case. For \(x=-5, y=3\) in \(x^{2}-y^{2}\), it becomes \((-5)^{2} - 3^{2}\).

Step 4: Evaluation for Case (b)

Again, following PEMDAS/BODMAS, squares come first. Calculate -5 squared (25) and 3 squared (9) separately. Then subtract these values: \(25 - 9 = 16\). So, for \(x=-5, y=3\), the expression \(x^{2} - y^{2}\) evaluates to 16.

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables with their corresponding numerical values. For instance, in the exercise given, we have the expression \(x^2 - y^2\). To evaluate this expression, we need to know the values of \(x\) and \(y\).

When \(x=4\) and \(y=3\), the substitution process involves replacing \(x\) with 4 and \(y\) with 3 in the expression. That is how we get \(4^2 - 3^2\). This simplifies our problem to dealing with numbers instead of variables, making the algebraic expression easier to evaluate.

It's essential to pay close attention to signs while substituting. In the case of \(x=-5\), substituting \(x\) with -5 means we'll have \( (-5)^2 - 3^2 \) which is not the same as \( -5^2 - 3^2 \). Parentheses are crucial here to avoid a common mistake: misinterpreting the negative sign and squaring the number incorrectly.

PEMDAS/BODMAS

The PEMDAS/BODMAS rule is a mnemonic device that helps students remember the order of operations in mathematics: Parentheses/Brackets, Exponents/Orders, Multiplication/Division, and Addition/Subtraction.

When evaluating algebraic expressions like \(x^2 - y^2\), it is crucial to follow this order. We first handle any operations inside parentheses, then calculate exponents or powers, followed by multiplication or division from left to right, and lastly, perform any addition or subtraction from left to right.

Applying PEMDAS/BODMAS to the Exercise

In both case (a) and case (b), we calculate exponents first, \(4^2\) and \(3^2\) or \( (-5)^2 \) and \(3^2\), before subtracting the second square from the first. This method ensures we arrive at the correct result, which is especially important when dealing with negative numbers as in case (b).

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making them as straightforward as possible.

This involves combining like terms, reducing fractions, and applying arithmetic operations correctly. Simplification might not always result in a single number; sometimes it just makes the expression easier to understand or further manipulate mathematically.

Why Simplification Matters

Simplification reduces the complexity of algebra problems. It can help students identify the structure of an expression, which is critical when moving on to more advanced topics like factoring or solving equations. In our exercise, simplification means executing the subtraction \(16 - 9\) or \(25 - 9\) after the exponents have been calculated. It's a clean finish to a two-step process: substitute, then simplify.

Difference of Squares

The 'difference of squares' is a specific type of algebraic expression that takes the form \(a^2 - b^2\). It is unique because it can be factored into \(a+b)(a-b)\). The expression in our exercise, \(x^2 - y^2\), is a classic example of a difference of squares.

Recognizing the difference of squares is valuable because it offers a shortcut to simplification and solution of more complicated problems.

Practical Application

Although our exercise does not require factoring, understanding that \(x^2 - y^2\) can be represented as \( (x+y)(x-y) \) will be useful for more complex algebra tasks where you may be asked to factor expressions, solve quadratic equations, or further simplify algebraic fractions.