Problem 49 - Fundamentals of Mathematics

Question

For problems \(47-56\), simplify each expression by combining like terms. $$ -7 b+4 m-3+3 n $$

Step-by-Step Solution

Answer

The simplified expression is \(-7b + 4m - 3 + 3n\).

Step 1: Identify Like Terms

First, let's identify terms that can be combined. In the expression \(-7b + 4m - 3 + 3n\), each term is already grouped by its unique variable or lack thereof. None of these terms can be grouped together because the variables differ: \(b\), \(m\), \(n\), and the constant \(-3\).

Step 2: Analyze the Expression

There are four types of terms: - The \(-7b\) term includes the variable \(b\).- The \(4m\) term includes the variable \(m\).- The \(3n\) term includes the variable \(n\).- The constant term, \(-3\) , has no variables attached to it.

Step 3: Combine Like Terms

Since there are no like terms to combine from our analysis above, the expression simplifies to itself:\(-7b + 4m - 3 + 3n\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in simplifying algebraic expressions. It involves identifying terms within an expression that have the same variables raised to the same power. Let's break this down to understand better.

In the expression \(-7b + 4m - 3 + 3n\), each term is different because they involve different variables or are constants. To combine like terms, we look for terms that have the same variable. Here's how you can approach it:

• Look for terms that involve the same variable. For example, terms with the variable \(b\) can be combined with other terms that also include \(b\).
• Ensure that the power of the variables is the same. In this context, you can combine \(b^2\) only with other \(b^2\) terms, not with \(b\) or \(b^3\).
• Combine constants separately since they do not include any variables.

This expression demonstrates a case where no terms are "like" because each variable is different or none (as with constant terms). Hence, the expression \(-7b + 4m - 3 + 3n\) remains unchanged after attempting to combine like terms.

Variables

Variables are symbols or letters that represent unknown values in expressions and equations. They allow us to generalize mathematical concepts and formulate equations that apply to various scenarios.

Take the expression \(-7b + 4m - 3 + 3n\):

• The term \(-7b\) contains the variable \(b\). Here, \(b\) is a placeholder for a potential value that can change the expression's outcome.
• The term \(4m\) involves the variable \(m\). Like \(b\), \(m\) can also represent different values depending on the context.
• The term \(3n\) involves the variable \(n\).

Variables are essential because they allow flexibility and are the foundation for creating algebraic expressions. They can represent numbers, objects, or datasets in mathematical models.

Constant Terms

Constant terms are components of an algebraic expression that do not include variables. Unlike variable terms, which can change based on the variable's value, constant terms stay the same regardless of the variables.

In the expression \(-7b + 4m - 3 + 3n\), there is one constant term:

• The number \(-3\) is a constant because it does not depend on any variable's value. It remains \(-3\) whether \(b\), \(m\), or \(n\) change.

Constant terms become pivotal in algebraic operations and equations as they provide a fixed value within calculations. Understanding constants helps simplify expressions and solve equations efficiently.