Problem 34 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. three decreased by a number

Step-by-Step Solution

Answer

The algebraic expression for the phrase 'three decreased by a number' is \(3 - x\).

Step 1: Identify the variable

According to the problem, the variable \(x\) represents 'the number'. So wherever the phrase mentions 'a number', we should substitute it with \(x\).

Step 2: Translate words to operations

The phrase 'three decreased by a number' suggests subtraction. In mathematics, ‘decreased by’ is translated to a subtraction operation. Therefore, the statement can be rewritten as 'three subtract a number'.

Step 3: Write the Algebraic expression

When putting the variable and operation together, the phrase 'three decreased by a number' would be written in algebraic form as \(3 - x\).

Key Concepts

Variable Representation

In the world of algebra, a variable can be thought of as a symbol that stands in for an unknown or changing value. In this exercise, the variable is denoted by the letter \( x \). This \( x \) is representing 'a number', which we do not yet know.

Using variables allows us to write expressions that can hold true for various values. It's like creating a formula or a blueprint that can be filled with different numbers and still maintain its structure. This is especially useful in problems where we wish to find one or more solutions.

Here, whenever the exercise refers to 'a number', we use the variable \( x \) to represent it. This simplifies the statement and allows us to work through the problem algebraically.

Phrase Translation

Phrase translation in algebra involves converting a verbal expression into a mathematical one. This is a crucial skill as it allows us to solve problems systematically. In this example, the English phrase given is 'three decreased by a number'.

To translate, use the operation indicated by the words. The key term here is 'decreased by'. In mathematical terms, this means subtraction. Our task is to change this phrase into an algebraic expression.

• 'Three' becomes the number \( 3 \).
• 'Decreased by' indicates subtraction.
• 'A number' is represented by the variable \( x \).

With these translations, the phrase becomes the expression \( 3 - x \). Understanding this translation process is very important to accurately solve problems in algebra.

Subtraction Operation

The subtraction operation is one of the basic functions in mathematics that is represented using the minus sign (\(-\)). Understanding how subtraction works in algebra is fundamental to manipulating algebraic expressions.

In our exercise, the term 'three decreased by a number' directly leads to a subtraction operation. Let's break it down:

• 'Three' is our starting point or the minuend.
• The term 'decreased by' indicates that something is being taken away.
• Here, 'a number' refers to our placeholder variable \( x \), which is being subtracted from three.

This boils down to the expression \( 3 - x \), where \( 3 \) is reduced by \( x \). Recognizing words like 'decreased by' is a cue to apply subtraction, helping transform verbal statements into useful algebraic equations.