Problem 80 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$5-7 b-13-4 b$$

Step-by-Step Solution

Answer

The simplified version of the algebraic expression is \(-11b - 8\).

Step 1: Identify Like Terms

In the given algebraic expression \(5-7b-13-4b\), the like terms are -7b and -4b (these are called variable terms), 5 and -13 (these are called constant terms).

Step 2: Combine Like Terms

Combine the variable terms: \(-7b - 4b = -11b\). Combine the constant terms: \(5 - 13 = -8\).

Step 3: Write Final Expression

The simplified expression should be presented as the sum of the simplified variable term and the simplified constant term, hence: \(-11b - 8\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra, allowing you to rewrite mathematical statements in a simpler form. The key is to reduce the expression without changing its value. Simplifying expressions mainly involves removing unnecessary complexity by combining like terms, which we'll explore further in the next sections.
Algebraic expressions can include variable terms (terms with variables like \(b\) in the expression) and constant terms (numbers without variables). The goal is to neatly combine these terms to reach a more compact version of the original statement.
By regularly practicing simplification, you'll find it easier to work with algebraic expressions, which will prove essential in solving equations and inequalities.

Combining Like Terms

Combining like terms is one of the foundational steps in simplifying expressions. But what exactly are 'like terms'? These are terms within an expression that share the same variables raised to the same power. For instance, in the expression you have, both \(-7b\) and \(-4b\) are like terms because they each include the variable \(b\).
Here's how you combine like terms:

• Add or subtract the coefficients of the like terms.
• Keep the common variable part as it is.

For example, combining \(-7b\) and \(-4b\), you add the coefficients to get \(-7 - 4 = -11\), resulting in \(-11b\). This step simplifies multiple similar variable terms into one, making the expression simpler and easier to understand.

Variable Terms

Variable terms are components of an expression that include a variable, such as \(-7b\) and \(-4b\) in the given exercise. The key part of a variable term is the variable, which could be any letter like \(x\), \(y\), or \(b\), representing unknown or changeable values.
Here is what makes up a variable term:

• The coefficient: This is the number that multiplies the variable (e.g., -7 in \(-7b\)).
• The variable itself: The letter that indicates the unknown or variable quantity (e.g., \(b\)).

When simplifying expressions, you focus on the coefficients of these terms when combining them. Simplification doesn't change the letters (variables), just how they're represented.

Constant Terms

Constant terms in an expression are those that do not contain any variables, thereby maintaining a fixed value. In the expression provided, these are 5 and \(-13\). Since they don't change within the context of the expression, they can be directly added or subtracted as regular numbers.
Here's what you need to know about constant terms:

• They are standalone numbers without any accompanying letters.
• When simplifying, simply perform basic arithmetic operations on constants.

In the exercise, you perform the operation \(5 - 13\), resulting in \(-8\). By handling constant terms separately from variable terms, you can simplify expressions efficiently and accurately.