Problem 84 - Algebra 2

Question

Evaluate \(\frac{a}{1-b}\) for the given values of \(a\) and \(b\) $$ a=-1, b=0.5 $$

Step-by-Step Solution

Answer

The value is -2.

Step 1: Identify the Formula

The expression given is \( \frac{a}{1-b} \). Our task is to substitute the values of \(a\) and \(b\) into this expression.

Step 2: Substitute Values

Substitute \( a = -1 \) and \( b = 0.5 \) into the expression. This gives us \( \frac{-1}{1-0.5} \).

Step 3: Simplify the Denominator

Calculate \( 1 - 0.5 \), which simplifies to \( 0.5 \).

Step 4: Evaluate the Expression

With the calculation from the previous step, we substitute back into the expression, which is now \( \frac{-1}{0.5} \). Simplifying this gives us \( -2 \) since dividing by 0.5 is the same as multiplying by 2.

Key Concepts

Substitution

Substitution is a key concept in algebra that involves replacing variables in an algebraic expression with their given values. This allows us to evaluate the expression or solve equations easily. In our exercise, we are given the expression \( \frac{a}{1-b} \) and the values \( a = -1 \) and \( b = 0.5 \). By substituting these values into the expression, we change its form to \( \frac{-1}{1-0.5} \). Substitution is performed step-by-step:

• First, identify each variable in the expression.
• Next, replace each variable with the specified number.

This helps in reducing complex expressions into simple numerical ones that are easier to handle. It's an essential skill for solving a variety of math problems.

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to understand and work with by reducing it to its simplest form. In our example, after substitution, we have the fractional expression \( \frac{-1}{1-0.5} \). To simplify:

• First, perform operations in the denominator: \( 1 - 0.5 \), which results in \( 0.5 \).
• Next, complete the division: \( \frac{-1}{0.5} \), which simplifies the expression to \( -2 \).

Reducing fractions like \( \frac{-1}{0.5} \) means finding a simpler equivalent value, often transforming the problem by considering multiplication (since dividing by 0.5 is the equivalent of multiplying by 2). Simplification enhances clarity and makes further calculations much easier.

Rational Expressions

A rational expression is an algebraic expression that can be written as a fraction where both the numerator and the denominator are polynomials. These can look like common fractions, but instead of integers, they involve algebraic terms. In the given problem, \( \frac{a}{1-b} \) is a rational expression. Key characteristics include:

• The numerator and denominator may include variables like \( a \) or \( b \), numbers, and algebraic operations such as addition and subtraction.
• The denominator cannot be zero, as division by zero is undefined and results in invalid expressions.

Understanding rational expressions helps in seamlessly evaluating and simplifying them, which involves careful attention to any restrictions, like excluding any values that might lead to a zero denominator. This ensures that the expression is always valid for evaluation.