Problem 69 - Introductory and Intermediate Algebra for College Students 4th

Question

Identify the terms in each algebraic expression. $$-3 x-8 y$$

Step-by-Step Solution

Answer

The terms of the given algebraic expression \(-3x -8y\) are \(-3x\) and \(-8y\). Each term includes a coefficient and the corresponding variable

Step 1: Identify The First Term

The first term of the algebraic expression is \(-3x\). This term is made up of a numerical coefficient, \(-3\), and a variable, \(x\).

Step 2: Identify The Second Term

The second term of the algebraic expression is \(-8y\). Similarly to the first term, this term consists of a numerical coefficient, \(-8\), and a variable, \(y\).

Key Concepts

Terms in Algebra

In algebra, an expression is comprised of parts known as terms. Each term can be a single number, a variable, or numbers and variables multiplied together. For instance, in the expression \( -3x - 8y \), there are two distinct terms, which are separated by a minus sign.

It's important to recognize each term because they are the building blocks of algebraic expressions. When identifying the terms in an expression, look for plus or minus signs; these typically indicate separation between them. In the given exercise, the terms are individual entities that can be manipulated for various operations, like simplification or solving equations.

• The first term is \( -3x \) which tells us that the variable \( x \) is being multiplied by \( -3 \).
• The second term is \( -8y \) which shows the variable \( y \) multiplied by \( -8 \).

Understanding how to identify and work with individual terms is crucial for mastering algebraic operations and further algebraic concepts.

Numerical Coefficient

A numerical coefficient is a number used to multiply a variable within an algebraic term. It quantitatively describes how many times a variable is included. Intellectually it's similar to the idea of how many 'units' of something we have. A coefficient could be positive, negative, or any real number.

In the problem \( -3x - 8y \), the numerical coefficients are \( -3 \) and \( -8 \) for the variables \( x \) and \( y \) respectively.

• The term \( -3x \) has a numerical coefficient of \( -3 \) indicating there are three units of \( x \) taken away (because of the negative sign).
• Similarly, the term \( -8y \) has a numerical coefficient of \( -8 \) signifying that eight units of \( y \) are being subtracted.

Grasping the concept of numerical coefficients is key to understanding the magnitude and direction in which the variables are being affected within an algebraic expression.

Variables in Algebra

In algebra, variables are symbols, often letters, which represent unknown values. They are foundational elements in algebraic expressions that can vary or change. Variables allow for general statements about relationships between quantities and are used to formulate algebraic equations that can be solved.

In the expression \( -3x - 8y \) from the given exercise, \( x \) and \( y \) are both variables. Each of these can represent any number, and the exact value they hold can change depending on the context of the problem.

• In the term \( -3x \), \( x \) could be any number, and its value in the expression would then be multiplied by the numerical coefficient \( -3 \) to find the term's value.
• The term \( -8y \) works the same way with the variable \( y \) potentially representing a variety of numbers.

Understanding variables is fundamental to solving algebraic equations and performing a wide range of mathematical operations. They serve as placeholders that allow algebraic expressions to be manipulated abstractly before substituting with specific values.