Problem 24 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. Eleven added to three times a number.

Step-by-Step Solution

Answer

The expression is \( 3x + 11 \).

Step 1: Identify the Unknown

Let the unknown number be represented by the variable \( x \). This variable will help us in forming the equation from the given phrase.

Step 2: Translate 'Three Times a Number'

The phrase 'three times a number' can be translated to the mathematical expression \( 3x \). This is because 'times' implies multiplication, so we multiply the variable \( x \) by 3.

Step 3: Translate 'Eleven Added to Three Times a Number'

Now, we need to add eleven to the expression \( 3x \). This means we take the expression from the previous step and write: \( 3x + 11 \).

Key Concepts

Mathematical Expressions

Mathematical expressions are fundamental components in the field of mathematics that allow us to convey operations, relationships, and quantities in a clear and concise manner. They are made up of numbers, variables, and operation symbols such as addition, subtraction, multiplication, and division. Expressions do not contain an equal sign (unlike equations) and therefore do not solve to a particular value unless each variable is substituted with specific numbers.
For example, in the phrase "three times a number," the mathematical expression is written as \(3x\). Here, "3" represents a constant multiplier, "times" indicates multiplication, and "x" is a variable that stands for the unknown number.
Translating verbal phrases into mathematical expressions is a critical skill that is used to formulate problems into a language that can be manipulated and solved mathematically.

Equations

Equations are statements that assert the equality of two expressions. They are distinguished from expressions by the presence of an equal sign \(=\). Equations are used to determine the value of unknown variables that satisfy the condition of equality.
For instance, when the task is to translate a verbal sentence into an equation, one must identify the results expected from the statement. In our exercise, the phrase "eleven added to three times a number" leads to the expression \(3x + 11\). However, it becomes an equation only when set equal to a particular value. For a complete equation, it might look like \(3x + 11 = y\), where "y" represents the outcome after performing the operations indicated in the expression.
Understanding how to transition from expressions to equations involves recognizing how the components relate through the concept of equality and what specific conditions—often given in problems—need to be satisfied.

Algebraic Translation

Algebraic translation is the process of converting a verbal statement or phrase into a mathematical expression or equation. This translation is essential in algebra because it enables one to analyze and solve real-world problems mathematically by breaking them down into understandable parts.
The steps involved typically include identifying key mathematical terms in the phrase, recognizing patterns, and systematically converting each part into algebraic symbols. For example, in the phrase "eleven added to three times a number," key terms such as "added" suggest addition, "three times" implies multiplication by 3, and "a number" indicates an unknown value or variable.
Mastering algebraic translation involves familiarity with mathematical terminology and practice. This skill forms the foundation for solving increasingly complex algebraic problems, as it allows students to create a mathematically accurate representation of a problem from a purely verbal description.