Problem 15 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number plus seven.

Step-by-Step Solution

Answer

The mathematical expression is \( x + 7 \).

Step 1: Identify the Number

In this phrase, we refer to "a number" which is an unknown value often represented by a variable in mathematical expressions. Let's denote this unknown number as \( x \).

Step 2: Identify the Operation

The phrase contains the operation "plus," which indicates addition. Therefore, we will be adding the number seven to our variable.

Step 3: Construct the Expression

Now that we have the variable \( x \) for the number and have identified the operation as addition, we can construct the expression. A number plus seven can be represented as \( x + 7 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics that help us describe relationships between numbers, variables, and operations. An expression consists of one or more terms, which can include variables, constants, and mathematical operations like addition or subtraction. For example, the expression \( x + 7 \) is an algebraic expression. Here, \( x \) stands for a number that we don't know yet, and "7" is a constant. Expressions don't have an equals sign or any comparison operators, so they aren't complete equations. Instead, they allow us to concisely represent relationships and calculations. Understanding how to translate verbal phrases like "a number plus seven" into an algebraic expression is essential for solving mathematical problems, especially in more complex situations. It's like learning a new language where words become symbols, enabling us to work with and manipulate numbers effortlessly.

Mathematical Operations

Mathematical operations are the actions we perform on numbers or variables to obtain results. The basic operations are addition, subtraction, multiplication, and division. Each operation has specific terms associated with it. For instance:

• Addition is indicated by words like "plus," "sum," or "increased by."
• Subtraction uses "minus," "difference," or "decreased by."
• Multiplication is associated with "times," "product," or "multiplied by."
• Division corresponds to "divided by," "quotient," or "over."

In our exercise, the word "plus" signals the addition operation. Recognizing these key terms helps us translate sentences and phrases into mathematical expressions. As we encounter phrases in algebra, identifying and applying the correct operation is crucial for constructing accurate expressions. With practice, these operations become second nature, allowing you to effortlessly convert verbal problems into numerical solutions.

Variables in Mathematics

Variables are symbols or letters that stand in for unknown values or changing quantities in mathematics. They are pivotal in forming algebraic expressions and are commonly represented by letters such as \( x \), \( y \), or \( z \). In our example, the phrase "a number" suggests a variable because it's not specified; it's any number.Variables are used to generalize problems, making it easier to work with various numbers and scenarios without rewriting each specific case. In the expression \( x + 7 \), \( x \) is the variable. This not only allows flexibility by letting \( x \) assume different values but also forms the basis of algebraic thinking and problem-solving. Understanding how to work with variables opens the door to advanced topics in algebra and other mathematical disciplines. They help us set up equations, represent functions, and define systems to unravel more complex problems.