Problem 28 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When fourteen is added to two times a number the result is six.

Step-by-Step Solution

Answer

Equation: \( 2x + 14 = 6 \).

Step 1: Understand the Phrase

Identify the unknown in the phrase, which is referred to as 'a number'. We will represent this unknown with the variable \( x \).

Step 2: Translate 'Two times a number'

Express 'two times a number' mathematically. Since 'a number' is represented by \( x \), 'two times a number' is written as \( 2x \).

Step 3: Translate 'Fourteen is added to...'

We need to add 14 to the expression \( 2x \). Mathematically, this is expressed as \( 2x + 14 \).

Step 4: Translate 'The result is six'

The phrase 'the result is six' is translated as the equation being equal to 6. Therefore, \( 2x + 14 = 6 \).

Step 5: Formulate the Complete Equation

Combine the translations from Steps 3 and 4 to write the complete equation: \( 2x + 14 = 6 \).

Key Concepts

Unknown Variable

When we talk about an "unknown variable," we are referring to an element in a problem that we do not know yet. In algebra, this unknown is usually denoted by letters such as \( x \), \( y \), or \( z \). By assigning a variable to the unknown number, we can perform mathematical operations to uncover its value.

To solve problems involving unknown variables, follow these simple steps:

• Identify what you don't know in the problem statement.
• Choose a variable (usually \( x \)) to represent this unknown.
• Use this variable consistently in mathematical expressions or equations you create.

When we assign a variable like \( x \) to the unknown, we create an opportunity to translate words into mathematics, making it possible to solve for the unknown.

Mathematical Equation

A mathematical equation is a statement that two expressions are equal. It's often represented by an '=' sign between two algebraic expressions. For those learning algebra, understanding how these equations represent real-world problems is essential. In the exercise, we translated the phrase into the equation \( 2x + 14 = 6 \).

Here's how we build a mathematical equation:

• Translate phrases from word problems into mathematical symbols.
• Ensure that each side of the equation represents equal value.
• Incorporate operations such as addition or multiplication as represented in the problem statement.

This process of forming equations from words and ensuring both sides balance allows us to methodically solve for the unknown variable.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations like addition or multiplication. These expressions do not include an equality sign, unlike equations. In our example, we translated pieces of a phrase into the expression \( 2x + 14 \).

To form an algebraic expression from a word problem:

• Break down the problem into smaller segments.
• Translate each part of the segment into its mathematical counterpart.
• Combine these mathematical segments to form a coherent expression.

Remember, algebraic expressions set the foundation for building equations by translating complex word problems into solvable mathematical language.

Mathematical Translation

Mathematical translation is the process of converting words from a problem statement into mathematical symbols. This step is crucial for solving word problems since it allows us to communicate the problem in numbers and operations. Our exercise involved translating "When fourteen is added to two times a number the result is six" into an equation.

To make a successful mathematical translation:

• Identify keywords and phrases that correspond to mathematical operations (e.g., 'added to', 'times', 'result is').
• Convert these phrases into symbols (e.g., '+', '×', '=' respectively).
• Consistently use variables for any unknown quantities mentioned.

This translation bridges the gap between verbal problems and algebraic solutions, making it easier to approach and solve mathematical challenges.