Problem 21 - Fundamentals of Mathematics

Question

Specify each term. $$-6 a-5 b$$

Step-by-Step Solution

Answer

Terms: \(-6a\) and \(-5b\).

Step 1: Identify Terms

Terms in an algebraic expression are individual components separated by plus (+) or minus (-) signs. Here, the expression is \(-6a - 5b\).

Step 2: Specify Each Term

In the expression \(-6a - 5b\), there are two terms. The first term is \(-6a\) and the second term is \(-5b\). Each term consists of a constant (coefficient) and a variable.

Step 3: Breakdown of the First Term

The first term \(-6a\) consists of a coefficient \(-6\) and a variable \(a\). This means \(-6a\) stands for \(-6\) times \(a\).

Step 4: Breakdown of the Second Term

The second term \(-5b\) consists of a coefficient \(-5\) and a variable \(b\). This means \(-5b\) stands for \(-5\) times \(b\).

Key Concepts

Coefficients

A coefficient is a number that multiplies a variable within an algebraic term. In the expression \(-6a - 5b\), the coefficients are \(-6\) and \(-5\). The sign of the coefficient indicates whether the term is positive or negative. Here are the main points to remember about coefficients:

• They determine the magnitude of the term. For instance, in \(-6a\), the coefficient \(-6\) shows how many times the variable \(a\) is counted.
• The coefficient can be positive or negative. If it is negative, as in \(-6\), it means that the term is subtracted.
• The absence of a written number in a term indicates a coefficient of 1. For example, the term \(x\) is essentially \(1x\).

Understanding coefficients is crucial because they affect how we perform operations with algebraic expressions. Remember, always combine like terms by their coefficients directly.

Variables

Variables are symbols used to represent unknown or changeable values in algebraic expressions. In the expression \(-6a - 5b\), the variables are \(a\) and \(b\). These symbols stand in for numbers, and here's more about them:

• Variables can take various values. Their exact value depends on the context or any given condition.
• They are crucial for creating formulas and expressions, allowing flexible usage in different scenarios.
• Variables are typically represented by letters such as \(x, y, a, b\), etc.

Variables might initially seem abstract, so think of them as placeholders for numbers. They allow you to generalize mathematical situations.

Terms in Algebra

In algebra, a term is a single mathematical expression. It can be a number, a variable, or a combination of numbers and variables, often involving coefficients and powers. Let's look into the expression \(-6a - 5b\), which includes two terms: \(-6a\) and \(-5b\). Here’s what you should know:

• Terms are the building blocks of algebraic expressions. They are separated by plus (+) or minus (−) signs.
• Each term consists of a coefficient and a variable factor, possibly raised to an exponent.
• The degree of a term is the exponent of the variable. If no exponent is shown, the variable is to the power of one.

Understanding how to identify and work with terms is essential for simplifying expressions and solving equations. Always focus on terms to break down and reorganize algebraic expressions effectively.