Problem 81 - Algebra 2

Question

Evaluate \(\frac{a}{1-b}\) for the given values of \(a\) and \(b\) $$ a=3, b=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The evaluated expression is 2.

Step 1: Identify the Formula

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

Step 2: Substitute the Values

Insert \( a = 3 \) and \( b = -\frac{1}{2} \) into the expression: \[ \frac{3}{1 - \left(-\frac{1}{2}\right)} \].

Step 3: Simplify the Denominator

Simplify the denominator \( 1 - (-\frac{1}{2}) \) to \( 1 + \frac{1}{2} \) which becomes \( \frac{3}{2} \).

Step 4: Evaluate the Expression

Now evaluate the expression: \[ \frac{3}{\frac{3}{2}} \]. This can be solved by multiplying 3 by the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \).

Step 5: Compute the Final Result

Multiply: \( 3 \cdot \frac{2}{3} \). This results in \( 2 \), as the threes cancel out.

Key Concepts

Substitution in Algebra

Substitution in algebra is a technique used to replace variables with specific values in an expression or equation. This method helps simplify and solve problems by plugging in known values for the variables.
Let's say you have an expression involving variables, like \( \frac{a}{1-b} \). You will substitute each variable with its given value. In our example:

• Substitute \( a = 3 \)
• Substitute \( b = -\frac{1}{2} \)

This becomes \( \frac{3}{1 - (-\frac{1}{2})} \).

This step is crucial for simplifying algebraic expressions and finding their value when the variables are known. It's like following a recipe: once you have the ingredients, you combine them as directed. Remember to always follow the correct order of operations to avoid mistakes.

Rational Expressions

Rational expressions are similar to fractions, but they consist of polynomials in the numerator and the denominator. Understanding these expressions involves knowing how to perform arithmetic operations such as addition, subtraction, multiplication, and division with them.
In our expression, \( \frac{a}{1-b} \) is a simple rational expression. The denominator here is \( 1 - b \), which requires careful simplification especially when involving negative values.
When substituting the value for \( b \), you get:

• In the denominator, \( 1 - (-\frac{1}{2}) \) changes to \( 1 + \frac{1}{2} \).
• This simplifies to \( \frac{3}{2} \).

Rational expressions often require simplification as part of their evaluation, and this can involve factoring, cancelling common terms, or simplifying the fraction itself. Keep in mind that division by zero is undefined, so you should always ensure the denominator does not equal zero.

Reciprocal Multiplication

Reciprocal multiplication is a key concept in simplifying complex fractions, particularly in cases involving division. The reciprocal of a number is simply one divided by that number. For example, the reciprocal of \( x \) is \( \frac{1}{x} \).

In solving our problem

• After substitution and simplification, our expression became: \( \frac{3}{\frac{3}{2}} \).
• To solve this, multiply 3 by the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \).

This multiplication: \( 3 \times \frac{2}{3} \) simplifies the expression, resulting in 2 as the threes cancel each other out.

Reciprocal multiplication can help streamline expression evaluation by turning division problems into multiplication problems. This method reduces the complexity in calculations and leads to a clearer understanding of how dividing by a fraction essentially works.