Problem 82 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression. Let \(x\) represent the unknown number. Five decreased by a number

Step-by-Step Solution

Answer

The algebraic expression is \( 5 - x \).

Step 1: Identify Key Words

The phrase "Five decreased by a number" can be broken into 'Five', 'decreased by', and 'a number'. We will translate each part into mathematical symbols.

Step 2: Translate 'Five' and 'a number'

In the given phrase, 'Five' is a constant number so it remains as 5. 'A number' refers to an unknown value which will be represented by the variable \( x \).

Step 3: Understand 'Decreased By'

The words 'decreased by' in mathematics generally indicate subtraction. Therefore, we need to subtract the unknown number from five.

Step 4: Formulate the Algebraic Expression

Putting together the translation from previous steps, 'Five decreased by a number' becomes \( 5 - x \).

Key Concepts

Translation of Phrases

In algebra, translating phrases into mathematical expressions is a crucial skill. It is the process of converting words into numbers and operations that create a mathematical statement. Let's break down how this is done in a simple example: the phrase 'Five decreased by a number.'
When translating:

• 'Five' is understood as the number 5.
• 'Decreased by' is a common math term meaning subtraction.
• 'A number' refers to an unknown quantity, typically represented by a variable such as \(x\).

By identifying these parts, we can reorganize the phrase to the algebraic expression \(5 - x\). This method allows students to better understand how language corresponds to mathematical operations.

Variables in Algebra

Variables are fundamental elements in algebra that represent unknown or changeable values. In the phrase 'Five decreased by a number,' 'a number' is the unknown part and is symbolized by a variable. Typically, variables are denoted by letters such as \(x\), \(y\), or \(z\).
Here’s why variables are important:

• They allow for the expression of general mathematical principles where specific numbers may be unknown.
• They help in forming equations or inequalities that are essential for solving problems.
• They enable you to describe situations and rules that are otherwise not possible to do with just numbers.

In this case, the expression \(5 - x\) uses \(x\) as a placeholder for any number, making it versatile for numerous scenarios.

Mathematical Operations

Mathematical operations are the foundation of forming and solving algebraic expressions. In the case of the phrase 'Five decreased by a number,' the operation involved is subtraction. Let's explore this in more detail:

• **Addition** adds two or more numbers to find their total sum. Phrases involving 'added to' or 'increased by' usually imply addition.
• **Subtraction** involves taking one number away from another. Terms like 'decreased by' or 'less than' typically mean subtraction, as seen in our example expression \(5 - x\).
• **Multiplication** and **division** are other key operations but are not used in this particular exercise. Phrases like 'product of' or 'times' require multiplication, while 'quotient of' suggests division.

Understanding these operations allows you to interpret and write algebraic expressions from verbal statements accurately, facilitating easier problem-solving in mathematics.