Problem 2 - Fundamentals of Mathematics

Question

Specify the terms in each expression. $$3 m-6 n$$

Step-by-Step Solution

Answer

Terms are \(3m\) and \(-6n\).

Step 1: Identify Coefficients

Look at the expression \(3m - 6n\). The first term is \(3m\) where 3 is the coefficient. The second term is \(-6n\) where -6 is the coefficient.

Step 2: Identify Variables

In the expression \(3m - 6n\), identify the variables: the first term \(3m\) has the variable \(m\), and the second term \(-6n\) has the variable \(n\).

Step 3: Specify the Terms

The expression consists of two terms: \(3m\) which is 3 times the variable \(m\), and \(-6n\) which is -6 times the variable \(n\).

Key Concepts

Coefficients

In algebra, coefficients are numbers that multiply the variables in an expression. Think of them as the 'partners' to the variables, showing how many times the variable is counted. For example, in the expression \(3m - 6n\), the numbers 3 and -6 are coefficients.

• The coefficient 3 in \(3m\) tells us that the variable \(m\) is taken 3 times.
• Similarly, the coefficient -6 in \(-6n\) tells us that \(n\) is taken -6 times, indicating subtraction or oppositeness in magnitude.

Understanding coefficients is crucial because they tell us about the scaling factor of each term in relation to the variable. A positive coefficient increases the size as the variable increases, while a negative one decreases it or implies a direction (like loans or debts in finance).

Variables

Variables in algebra are symbols that stand for unknown or changeable values. They help us generalize mathematical expressions, so we can apply them broadly across different scenarios. In the expression \(3m - 6n\), the letters \(m\) and \(n\) are our variables.

• \(m\): It can be any number, and the value may change depending on the problem.
• \(n\): Similarly, \(n\) can vary, representing different numbers.

Variables function as placeholders. They are essential in forming expressions that model real-world situations, like calculating areas or predicting trends. Identifying variables helps in solving equations since they are the elements you solve for. In this way, variables give flexibility and power to algebraic operations.

Terms in algebra

Terms in algebra are the building blocks of expressions. They consist of variables, coefficients, and possibly constants (which are just standalone numbers). In the expression \(3m - 6n\), there are two terms: \(3m\) and \(-6n\).

• \(3m\): This term is a combination of the coefficient 3 and the variable \(m\).
• \(-6n\): This term combines the coefficient -6 with the variable \(n\).

A term may have only one variable or multiple variables, and it could also be a constant with no variable part. Understanding terms is fundamental because simplifying expressions heavily relies on combining or breaking down terms. Terms make up equations and inequalities, and handling them correctly is key to solving algebraic problems. They allow us to efficiently organize and represent mathematical ideas.