Problem 86 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$12 b-13 b$$

Step-by-Step Solution

Answer

\(-b\)

Step 1: Identify Like Terms

In the expression \(12b - 13b\), both 12b and 13b are like terms because they have the same variable 'b'.

Step 2: Combine Like Terms

The like terms can be combined by subtracting the coefficients. Subtract 13 from 12 which results in -1. So, \(12b - 13b = -1b\).

Step 3: Simplify The Result

-1b is typically written as -b. Therefore, \(12b - 13b = -b\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental aspect of algebra that signify the combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. Think of them as a way to represent mathematical ideas using symbols. An expression can be as simple as a single number or variable, or as complex as an equation with multiple terms and operations. Each part of an algebraic expression that is separated by a plus or minus sign is referred to as a 'term'.

In the exercise provided, we see an algebraic expression: \(12b - 13b\). In this expression:

• 12b and 13b are the terms.
• The variable involved here is 'b'.
• The coefficients in the terms are the numbers in front of the variable, which are 12 and -13.

Understanding these components is the first step in working with and simplifying algebraic expressions. They allow us to perform operations systematically by following mathematical rules.

Combining Like Terms

Combining like terms is a crucial skill in simplifying algebraic expressions. Like terms are terms within an expression that have the same variable raised to the same power. The coefficients of these terms can be directly added or subtracted. This process simplifies an expression, making it easier to understand and solve.

In our example, both \(12b\) and \(13b\) are like terms. This is because they share the variable 'b' without any exponents or different powers. The coefficients are combined by completing the specified operation; in this case, subtracting. So, you subtract the coefficient 13 from 12, resulting in -1.

• It’s important to focus on the coefficients when dealing with like terms.
• Variables and their exponents must match for terms to be considered like terms.
• Combining like terms reduces the expression to fewer terms.

This step is pivotal to moving towards a simplified and more usable form of the expression.

Simplifying Expressions

Simplifying expressions involves breaking down the expression into its simplest form. This process essentially involves removing any unnecessary parts of the expression while maintaining its value and integrity. Simplification helps us make sense of complex expressions and solve equations more easily.

Taking the expression \(-1b\) from the previous step, simplification involves rewriting this in a more conventional form. Typically, mathematicians prefer expressions to have the simplest form which means -1b is simply written as \(-b\).

• Simplifying does not change the value of the expression, just the way it looks.
• The simplified form is generally preferred in mathematical communication.
• This stage is essential in solving equations as it provides clarity and conciseness.

Overall, simplifying algebraic expressions is not just about reducing clutter; it's about recognizing patterns and relationships within the expression to make it easier to interpret and use in various mathematical scenarios.