Problem 18 - Prealgebra

Question

Identify the terms, like terms, coefficients, and constants in each expression. \(6 m-2 n+7\)

Step-by-Step Solution

Answer

Terms: \(6m\), \(-2n\), \(7\); Like terms: None; Coefficients: \(6, -2\); Constant: \(7\).

Step 1: Identify the Terms

To identify the terms in the expression, we need to locate the parts separated by plus or minus signs. The given expression is \(6m - 2n + 7\). This expression has three terms: \(6m\), \(-2n\), and \(7\).

Step 2: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In this expression, \(6m\) and \(-2n\) are not like terms because they have different variables. The number \(7\) is a constant and doesn't have any like terms with the variables present.

Step 3: Identify the Coefficients

Coefficients are the numerical parts of terms that contain variables. In the expression, the term \(6m\) has a coefficient of \(6\), and the term \(-2n\) has a coefficient of \(-2\). These numbers indicate how many times the variable is multiplied.

Step 4: Identify the Constants

Constants are terms that do not contain any variables and are plain numbers. In this expression, \(7\) is the constant term as it is not attached to any variable.

Key Concepts

Terms in Algebra

Algebraic expressions are made up of individual elements known as terms. A term can be a number, a variable, or a combination of both multiplied together. In the case of the expression \(6m - 2n + 7\), terms are the distinct parts divided by the plus or minus signs. This helps us separate our expression into easier-to-manage pieces.

The terms in our example are:

• \(6m\)
• \(-2n\)
• \(+7\)

Each piece plays a unique role in forming the complete expression.

Like Terms

When working with algebraic expressions, identifying like terms is crucial. Like terms have the same variable components raised to the same power. This allows us to combine them more easily during calculations.

In the expression \(6m - 2n + 7\), due to the different variables of \(m\) and \(n\), there are no like terms with each other. Additionally, the constant \(7\) cannot pair with any term because it lacks a variable component. Here’s what you need to remember about like terms:

• Must have the same variable.
• Variables must be raised to the same power.
• Only like terms can be combined in simplifying expressions.

Coefficients

Coefficients are the numerical parts found in terms with variables. Understanding coefficients is key because they indicate how much of the variable is included in the expression. For example, they tell us how many times the variable should be considered.

In \(6m - 2n + 7\):

• The coefficient of \(6m\) is \(6\).
• The coefficient of \(-2n\) is \(-2\).

These numbers determine the impact of the variables within the whole expression.

Constants in Expressions

Constants are the fixed values within an algebraic expression. Unlike terms containing variables, constants maintain the same value across different scenarios and calculations.

In the expression \(6m - 2n + 7\),

• \(7\) is the constant term.

A constant doesn't fluctuate with the change in variable values, holding its ground as an unchanging number.