Problem 36 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. five times a number, decreased by 3

Step-by-Step Solution

Answer

The English phrase 'five times a number, decreased by 3' can be written as the algebraic expression \(5x - 3\).

Step 1: Translate 'Five Times a Number'

The phrase 'five times a number' translates into an algebraic operation of multiplying a number by five. Here, number is represented by the variable \(x\). So this part of the statement can be written as \(5x\).

Step 2: Translate 'decreased by 3'

'Decreased by 3' means that three is subtracted. It affects the result of the previous operation (five times a number). This translates to '-3' in our expression.

Step 3: Combine Both Parts

Combine both parts of the expressions. We are decreasing the result of the five times the number by three. So, our final algebraic expression is \(5x - 3\).

Key Concepts

Translating English to Algebra

Understanding how to convert everyday language into algebraic expressions is like learning a new language; it's a skill that can be improved with practice. The key is recognizing the math operations and quantities described in words and knowing how to represent them with algebraic symbols and numbers. For instance, when we hear 'five times a number', we should visualize that 'times' signals multiplication, and 'a number' can be any value, hence we use a variable, often represented as a letter like \(x\), \(y\), or \(n\).

Anytime you're faced with translating English phrases into algebraic terms, reflect on what the words imply mathematically. Phrases like 'decreased by', 'the sum of', 'product of', and 'divided by' give clues to the operations subtraction, addition, multiplication, and division, respectively. Let's apply this to our problem. 'Five times a number' immediately suggests taking our variable, \(x\), and performing multiplication: \(5x\). Then we have 'decreased by 3', implying subtraction from this product: \(5x - 3\). It’s a step toward creating a universal math language that can be understood by others versed in algebra.

Variables in Algebra

Variables are foundational elements in algebra that serve as placeholders for numbers whose values are not yet known or can change. They are usually represented by letters from the alphabet and can be included in algebraic expressions to describe a relation or perform a calculation regardless of the specific number they represent.

Let's imagine variables as empty boxes that can hold any number. In our example, we chose \(x\) to hold the place of 'a number'. It's important when solving algebraic expressions or equations to remain consistent with the chosen variable. This concept allows us to generalize problems, create formulas, and explore patterns. Whether you are adding \(x + 7\), multiplying \(3x\), or expressing a decrease such as \(x - 3\), the variable \(x\) remains a powerful tool in constructing mathematical statements that model real-life situations.

Writing Algebraic Expressions

When writing algebraic expressions, the goal is to create a symbolic representation of mathematical operations that are described either verbally or through real-world scenarios. These expressions are composed of variables, numbers, and arithmetic operations.

To write adeptly, one must understand the order of operations and proper placement of parentheses for clarifying the sequence of calculations. For example, if an expression states 'twice the sum of a number and three', writing \(2x + 3\) is incorrect. The correct form with parentheses is \(2(x + 3)\) which preserves the intended order of operations - sum first, then multiplication.

In reference to our exercise, 'five times a number, decreased by 3' requires translating into mathematical symbols order-wise. First, we write the multiplication \(5x\) and then implement the decrease, leading to the final expression: \(5x - 3\). This careful construction ensures that the algebraic expression accurately reflects the given phrase or situation and allows it to be universally understood and solved.