Problem 22 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(y=-3 .\) See Example 1. $$ \frac{-y-11}{y^{2}+2 y-3} $$

Step-by-Step Solution

Answer

The expression is undefined because the denominator equals zero.

Step 1: Substitute the value of y

Replace the variable \( y \) in the expression with \( -3 \). The expression becomes \( \frac{-(-3)-11}{(-3)^{2}+2(-3)-3} \).

Step 2: Simplify the numerator

Evaluate the numerator:\(-(-3) - 11 = 3 - 11 = -8\).

Step 3: Simplify the denominator

Calculate each term in the denominator:1. \((-3)^2 = 9\). 2. \(2(-3) = -6\). 3. \(-3\). Combine these values: \[9 - 6 - 3 = 0\].

Step 4: Evaluate the expression

The simplified expression is \( \frac{-8}{0} \), but division by zero is undefined.

Key Concepts

Division by Zero

Division by zero occurs when an expression's denominator is zero. In mathematics, this is considered undefined because dividing any number by zero does not yield a finite number or a meaningful result. If you encounter a situation where you might divide by zero, such as the expression \( \frac{-8}{0} \), it indicates an undefined result.

When solving problems involving fractions or rational expressions, always ensure the denominator is not zero. If an expression reaches a division by zero scenario, the calculation cannot proceed further without an error. This underscores the importance of clearly understanding expressions' compositions before attempting operations, like division. Also, remember: dividing by zero is a key consideration in real-world problem-solving and programming, where it can lead to critical errors.

Substitution in Expressions

Substitution is a useful technique in algebra that involves replacing variables in an expression with given numerical values. Here, we substitute \( y = -3 \) into the expression \( \frac{-y-11}{y^{2}+2y-3} \). This gives you the new expression \( \frac{-(-3)-11}{(-3)^{2}+2(-3)-3} \).

This step requires you to carefully replace each instance of the variable with its numerical equivalent to ensure accurate computations. Keep track of signs, particularly with negative numbers. Remember, replacing variables allows for solving equations and aiding in graphing functions, providing a clearer picture of the behavior of mathematical models.

Numerator Simplification

Simplifying the numerator helps to reduce the computational complexity of an expression. The numerator in our example \(-(-3) - 11\) can be solved as follows: First, the double negative becomes positive, transforming \(-(-3)\) into \(+3\).

Simplify further by computing \(3 - 11\), resulting in \(-8\). Breaking down expressions step-by-step ensures all details are addressed to prevent mistakes in further calculations. Simplifying the numerator often streamlines the evaluation process, leading to a more manageable expression.

Denominator Simplification

The denominator's simplification is crucial for evaluating expressions properly. In the equation \((-3)^2 + 2(-3) - 3\), evaluate each term separately.

• Calculate \((-3)^2\), which equals 9.
• Then, \(2(-3)\) becomes \(-6\).
• The constant term remains \(-3\).

Adding these results together: \(9 - 6 - 3 = 0\). Seeing that the denominator sums to zero is essential because it results in division by zero, concluding the expression is undefined. This process stresses how crucial simplifying expression parts is to grasp fully their potential outcomes. Always note potential undefined states when the denominator equals zero to avoid missteps.