Problem 21 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(y=-3 .\) See Example 1. $$ \frac{y^{2}+9}{9-y^{2}} $$

Step-by-Step Solution

Answer

The expression is undefined because division by zero is not allowed.

Step 1: Substitute the Value of y

Replace the variable \( y \) in the expression with \( -3 \). The expression becomes \( \frac{(-3)^{2} + 9}{9 - (-3)^2} \).

Step 2: Evaluate the Exponents

Calculate \((-3)^2\). This results in \(9\). Therefore, the expression is now \( \frac{9 + 9}{9 - 9} \).

Step 3: Simplify the Numerator

Add together the numbers in the numerator: \(9 + 9 = 18\). So the expression is \( \frac{18}{9 - 9} \).

Step 4: Simplify the Denominator

Calculate the denominator: \(9 - 9 = 0\). So the expression is \( \frac{18}{0} \).

Step 5: Evaluate Expression with Denominator Zero

An expression with a zero denominator is undefined in mathematics. Therefore, \( \frac{18}{0} \) is undefined.

Key Concepts

Evaluation of Expressions

When evaluating algebraic expressions, your goal is to determine their value by following specific operations. To evaluate an expression like \( \frac{y^2 + 9}{9 - y^2} \) for \( y = -3 \), you start by substituting the given value for the variable. After substitution, each component of the expression involving the variable must be calculated. For example, if you substitute \( y = -3 \) into the expression, you will first resolve any exponents: \((-3)^2 = 9\). This changes your expression to \( \frac{9 + 9}{9 - 9} \). Once any numerical operations like exponents or multiplications are resolved, the next step is to solve the arithmetic operations like addition or subtraction in both the numerator and the denominator. This results in the expression transforming to \( \frac{18}{0} \). To summarize, evaluating expressions requires:

• Substituting the variable with the given value.
• Solving exponents and multiplications.
• Simplifying the resulting operations to find the expression's value.

Understanding how to evaluate expressions is a fundamental skill in algebra.

Undefined Expressions

In algebra, an expression is said to be undefined if it results in a mathematical impossibility. One common situation is the division by zero. When you reach \( \frac{18}{0} \), you see that the denominator equals zero, making the whole expression undefined. Mathematically, division by zero does not produce a real number, so any expression or calculation with division by zero is "undefined." Thus, you cannot determine a specific value or number when an expression has a zero in the denominator, like \( \frac{a}{0} \) where \( a \) is any non-zero number.Remember these key points:

• Zero as a denominator always makes an expression undefined.
• An undefined expression means the calculation cannot be completed in normal mathematical terms.
• Division by zero is a fundamental impossibility in arithmetic and is crucial to recognize in algebraic expressions.

Recognizing undefined expressions helps avoid errors in calculations and enhances understanding of algebra.

Substitution in Algebra

Substitution is the method of replacing variables with numbers to simplify expressions and solve equations. This is especially useful when you have a known value for a variable, like \( y = -3 \) in our exercise.The substitution method transforms algebraic expressions into simpler numeric forms, allowing for straightforward calculations. You replace all instances of the variable with the given number, and then perform the usual arithmetic operations. For example, substitute \( y = -3 \) in \( \frac{y^2 + 9}{9 - y^2} \), changing the problem to \( \frac{(-3)^2 + 9}{9 - (-3)^2} \).To effectively use substitution, follow these steps:

• Identify the variable(s) in the expression.
• Replace each variable with the given numeric value.
• Perform arithmetic calculations step-by-step to simplify the expression.

Substitution is a direct application of given values in algebra, simplifying the process to make complex problems manageable.