Problem 70 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. the product of 5 and a number, which is then subtracted from the product of 11 and a number

Step-by-Step Solution

Answer

The English phrase translates to the algebra expression \(11x - 5x\), which simplifies to \(6x\).

Step 1: Translate the Phrase to Algebra

First, let's translate each part of the phrase in the order it is given. 'The product of 5 and a number' can be translated to \(5x\). 'Subtracted from the product of 11 and a number,' can be written as \(11x - 5x\). Therefore, the whole phrase translates to \(11x - 5x\).

Step 2: Simplify the Expression

Now, to simplify the expression, subtract \(5x\) from \(11x\). The result is \(6x\). This is the simplified form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing an expression to its simplest form. This means combining like terms to make the expression easier to work with. In our example, "the product of 5 and a number, which is then subtracted from the product of 11 and a number," is translated into the expression \(11x - 5x\).

When simplifying, the first step is to identify like terms. Like terms have the same variable raised to the same power. In the expression \(11x - 5x\), both terms have the variable \(x\). Since they are like terms, you can perform operations directly on their coefficients—the constants that multiply the variable.

To simplify \(11x - 5x\), you subtract 5 from 11, resulting in \(6x\). Simplifying expressions makes it easier for you to solve algebraic equations or understand relationships between variables.

Translating Phrases into Algebra

Translating verbal phrases into algebraic expressions is like learning a new language. You are converting words into mathematical symbols. Each word or phrase corresponds to a specific mathematical operation or expression. This process requires careful reading and understanding of the language used in the problem.

Let's take the example: "the product of 5 and a number, which is then subtracted from the product of 11 and a number." Here, "product" means multiplication. So, "the product of 5 and a number" becomes \(5x\). Similarly, "the product of 11 and a number" becomes \(11x\).

The key part is "subtracted from," which shows the order of operations: \(11x - 5x\). Always remember the importance of order, as it can change the meaning entirely—the word "from" indicates that \(11x\) comes first. Accurately translating phrases helps you form correct equations, which is essential for solving algebraic problems.

Introductory Algebra

Introductory algebra is the foundation upon which you'll build more advanced mathematical concepts. It involves understanding and working with variables, expressions, and equations. Variables are symbols like \(x\) that stand for unknown or variable amounts.

One of the first steps in algebra is learning how to handle expressions, such as \(5x\) or \(11x\), which involve operations like addition or subtraction of these terms. The aim is to manipulate them using rules that maintain equality or simplify calculations.

In the given problem, introductory algebra concepts are demonstrated by translating a verbal phrase into an expression, then simplifying that expression. This process is crucial because it transforms everyday language into a form that can be more easily analyzed mathematically. Understanding these basics helps in solving more complex problems in algebra and beyond.