Problem 19 - Fundamentals of Mathematics

Question

Specify each term. $$m-3 n-4 a+7 b$$

Step-by-Step Solution

Answer

Terms are \( m \), \( -3n \), \( -4a \), and \( 7b \).

Step 1: Identifying Terms

The expression given is \( m - 3n - 4a + 7b \). Each term is separated by a plus or minus sign. Let's identify each term: \( m \), \( -3n \), \( -4a \), and \( 7b \).

Step 2: Term 1: Single Variable Term

The term \( m \) is a variable term consisting of just a single variable \( m \), with an implicit coefficient of 1.

Step 3: Term 2: Negative Coefficient Term

The term \( -3n \) consists of a variable \( n \) with a coefficient of \(-3\).

Step 4: Term 3: Another Negative Coefficient Term

The term \( -4a \) is made up of a variable \( a \) with a coefficient of \(-4\).

Step 5: Term 4: Positive Coefficient Term

The last term \( 7b \) includes a variable \( b \) and has a coefficient of 7.

Key Concepts

Terms Identification

When working with algebraic expressions, it's crucial to understand what terms are and how to identify them. An expression is made up of one or more terms, which are parts of the expression separated by plus (+) or minus (−) signs. For example, in the expression \( m - 3n - 4a + 7b \), we can spot four distinct terms:

• \( m \)
• \( -3n \)
• \( -4a \)
• \( 7b \)

Each term can be a single number, a single variable, or a product of numbers and variables. Being able to identify terms is the first step in understanding the structure of any algebraic expression. This knowledge helps in simplifying, solving, and performing operations on the expressions.

Coefficients in Expressions

In algebraic expressions, coefficients play an important role. Coefficients are the numerical part of a term. They tell you how many times to multiply the variable by. Let's examine the role of coefficients in each of the terms from our expression \( m - 3n - 4a + 7b \):

• For the term \( m \), the coefficient is implicitly \( 1 \). This means the term is equivalent to \( 1m \).
• In \( -3n \), \(-3\) is the coefficient, showing that \( n \) is multiplied by \(-3\).
• For \( -4a \), the coefficient is \(-4\), which means \( a \) is multiplied by \(-4\).
• Finally, \( 7b \) has a coefficient of \( 7 \), indicating that \( b \) is multiplied by \( 7 \).

Recognizing these coefficients is essential for manipulating and simplifying expressions. Coefficients not only determine the magnitude of the term but also indicate the sign, which affects how terms are combined.

Variables in Expressions

Variables are fundamental components of algebraic expressions. A variable represents an unknown or changeable number, and it is usually denoted by a letter. In our expression \( m - 3n - 4a + 7b \), we find variables in each term:

• \( m \) is a variable standing alone in its term.
• \( n \) is the variable in the term \( -3n \).
• \( a \) is the variable found in \( -4a \).
• \( b \) is the variable present in \( 7b \).

Variables in expressions can be single characters or sometimes longer symbols when more specificity is needed. They allow expressions to generalize conditions and solutions across many values. Understanding the role of each variable helps in evaluating expressions and determining the relationships within equations.