Problem 72 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. nine times the product of 3 and a number

Step-by-Step Solution

Answer

The simplified algebraic expression is \(27x\).

Step 1: Identify the algebraic expression

In the phrase 'nine times the product of 3 and a number', the 'number' is represented by \(x\). So 'the product of 3 and a number' will be \(3x\). 'Nine times the product of 3 and a number' translates to 'nine times \(3x\)', which gives the expression \(9 \cdot 3x\).

Step 2: Simplify the algebraic expression

To simplify this expression, multiply 9 by 3x, which gives \(27x\).

Key Concepts

Simplifying Expressions

The act of simplifying an algebraic expression is a fundamental process in algebra, aimed at making the expression as elementary as possible without changing its value or meaning. This often involves several steps, such as combining like terms, reducing fractions, or, as in our example, executing multiplication.

To simplify an expression like \(9 \cdot 3x\), which signifies 'nine times the product of 3 and a number', we combine the constants (9 and 3) through multiplication. The resulting product \(27\) then remains multiplied by the variable \(x\), leading to the simplified expression \(27x\). It's crucial to remember that simplifying an expression is about making it easier to understand and work with, not solving for a variable.

Translating English to Algebra

Translating English phrases into algebraic expressions is like turning words into mathematical codes. This skill enables students to interpret and solve real-world problems mathematically. For instance, when encountering a phrase like 'nine times the product of 3 and a number,' it is important to recognize the operation indicators: 'times' implies multiplication, while 'product' indicates that two quantities need to be multiplied together before applying the 'nine times' part.

Start with identifying and translating the core parts: 'a number' can be represented by any variable, commonly \(x\). Then, identify the 'product' as \(3x\). Lastly, multiply this product by nine to form the algebraic expression \(9 \cdot 3x\). Effective translation requires familiarity with operation keywords and practicing with various phrases to build confidence in this mathematical language.

Variable Representation

Variables in algebra act as placeholders for unknown or changeable values. By convention, letters like \(x\), \(y\), and \(z\) are commonly used, but any symbol can serve as a variable. In our exercise, \(x\) 'represents the number', allowing the phrase to describe a range of possibilities – any value of \(x\) will satisfy the general form of the expression.

Understanding the role of variables is crucial. In variable representation, one must distinguish between different quantities and their relationships. As you practice, it's essential to recognize patterns in how English phrases correspond to algebraic expressions, taking special care with terms that describe operations like addition, subtraction, multiplication, and division. Appropriately representing variables lays the foundation for constructing, simplifying, and solving algebraic expressions.