Problem 17 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number decreased by ten.

Step-by-Step Solution

Answer

The expression is \( x - 10 \).

Step 1: Identify the Key Components of the Phrase

The phrase given is 'A number decreased by ten.' Here, 'a number' represents an unknown value in mathematics. We often use a variable, like \( x \), to denote this unknown number.

Step 2: Interpret the Mathematical Operation

The words 'decreased by' indicate a subtraction operation. Hence, we are asked to subtract ten from the variable that represents our number.

Step 3: Formulate the Mathematical Expression

Based on the analysis above, the number decreased by ten is expressed as \( x - 10 \), where \( x \) is the variable representing the unknown number.

Key Concepts

Variables in Algebra

In algebra, variables are essentially symbols that stand for unknown values. They allow us to work with numbers even when we don't know exactly what those numbers are. This concept is fundamental, especially when dealing with algebraic expressions and equations.

• Definition: A variable is typically a letter like \( x \), \( y \), or \( z \) used to represent an unknown number.
• Purpose: Variables enable us to generalize mathematical concepts and solve problems for a range of possible solutions.

Using variables, such as \( x \), provides flexibility and allows us to formulate expressions that can express mathematical relationships in a broad and general way.

Subtraction Operation

Subtraction is one of the fundamental operations in mathematics. It allows us to find the difference between numbers and quantities.

• Definition: Subtraction is indicated by the "-" symbol and involves taking away one quantity from another.
• Key Words: Words such as "decreased by," "minus," "less," and "subtracted from" often indicate subtraction.
• Application: In the example of "a number decreased by ten," the word "decreased" tells us to use subtraction. Hence, we use \( x - 10 \) to indicate ten is subtracted from a number \( x \).

Understanding the language of subtraction is crucial in correctly translating real-world phrases into mathematical expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like addition or subtraction) that represent a particular quantity or concept.

• Elements: Expressions can include numbers, variables, and operations such as addition and subtraction.
• Purpose: They are used to convey relationships and computations clearly and concisely in a mathematical form.
• Example: The phrase "a number decreased by ten" translates to the expression \( x - 10 \). Here, \( x \) is a variable representing the unknown number, and "- 10" shows subtraction.

Crafting mathematical expressions accurately is key in solving and understanding algebraic problems.