Problem 90 - Introductory and Intermediate Algebra for College Students 4th

Question

Writing about mathematics will help you to learn mathematics. For all writing exercises in this book, use complete sentences to respond to the questions. Some writing exercises can be answered in a sentence; others require a paragraph or two. You can decide how much you need to write as long as your writing clearly and directly answers the question in the exercise. Standard references such as a dictionary and a thesaurus should be helpful. What is an algebraic expression?

Step-by-Step Solution

Answer

An algebraic expression is a mathematical phrase that can include numbers, operators (add, subtract, multiply, etc.), and at least one variable. Examples of algebraic expressions are 2x+3, 3(a+b), and 4xy. Even a single number, like 5, can be considered an algebraic expression.

Step 1: Understanding the term

Start by understanding the term 'algebraic expression'. An algebraic expression is a mathematical phrase that can contain ordinary numbers, 'variables' (like x or y) and operators (like add, subtract, multiply, etc). For example, 2x+3 is an algebraic expression.

Step 2: Providing examples

Now, provide a few examples to further clarify the concept. For example: \n1. 3(a+b) - This is an algebraic expression because it contains variables a and b, operator of addition within the parenthesis and multiplication outside.\n2. 4xy - This too is an algebraic expression having variables x and y with multiplication as the operator.\n3. 5 - This too can be considered an algebraic expression though it has no variable nor operator, it is a constant algebraic expression.

Key Concepts

Variables in Algebra

In algebra, the use of variables is fundamental as they represent unknown values that we aim to find. Variables are often denoted by letters like x, y, or z, and they can stand in for any number. This freedom allows us to create general solutions to problems that can be applied in multiple scenarios.

For instance, consider the equation 2x = 10. Here, x is a variable. By solving the equation, we find that x equals 5. Variables are not just placeholders; they are essential in describing relationships between quantities and in formulating equations that model real-world situations.

Why Use Variables?

• Generality: Variables allow us to write a formula that is applicable in many cases, not just for specific numbers.
• Solving problems: By using variables, one can set up and solve equations to find unknowns in a structured manner.
• Function description: Variables are used in functions to denote independent and dependent quantities, reflecting changes in one based on the other.

Mathematical Operators

Mathematical operators are symbols or words that signify computations. These include the fundamental operations of addition (+), subtraction (-), multiplication (× or ·), and division (÷ or /). They are the building blocks for constructing mathematical statements and for performing algebraic manipulations.

In an algebraic expression, operators tell us what to do with the variables and numbers involved. For example, in the expression 4x + 7, the + sign is an operator that instructs us to add 4 times the value of x to 7. Operators go beyond just these basics, including exponents (^), roots, and more advanced concepts such as logarithms which have their own set of rules.

Order of Operations

When multiple operators are present, the order in which the operations should be carried out is determined by the rules of precedence, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). This ensures that everyone can read and solve the expression in the same way.

Writing in Mathematics

Writing in mathematics involves clearly expressing ideas using numerical expressions, variables, operators, and a comprehensive set of symbols that form the language of mathematics. It's not just about finding the right answer, but also about explaining the process and reasoning behind mathematical problem-solving.

Good mathematical writing conveys complex ideas simply. It uses concise terms and clear notation, making the logical flow easy to follow. For example, when writing about the solution to an equation or explaining an algebraic expression like 3(a + b), one should not only state the final result but also describe the steps and operations that lead to it. This will typically include explaining why each operation is performed and how the variables are manipulated through each step of the problem.

Writing mathematically isn't limited to numbers and equations—it also includes the use of words to justify, hypothesize, describe patterns, and make arguments. It is a skill that enhances understanding and aids in the deeper exploration of mathematical concepts.