Problem 86 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression. Let \(x\) represent the unknown number. Twice a number, decreased by 72

Step-by-Step Solution

Answer

The expression is \(2x - 72\).

Step 1: Identify the parts of the phrase

The phrase "Twice a number, decreased by 72" can be broken down into two parts: "Twice a number" and "decreased by 72".

Step 2: Represent 'Twice a number' using algebra

To represent "Twice a number", we multiply the unknown number by 2. If the unknown number is denoted by \(x\), then "Twice a number" is represented as \(2x\).

Step 3: Interpret 'Decreased by 72'

The phrase "decreased by 72" indicates subtraction of 72 from the expression obtained in the previous step.

Step 4: Form the final algebraic expression

Subtract 72 from \(2x\) to form the complete expression. Thus, the final algebraic expression is \(2x - 72\).

Key Concepts

Algebra

Algebra is the branch of mathematics that helps us express mathematical relationships using symbols and letters. This makes it easier to solve problems even when we don't know the value of some numbers. Instead of working with specific numbers all the time, algebra allows us to use general expressions. By learning algebraic expressions, we can solve problems much more efficiently.

In our exercise, we were asked to convert a verbal phrase into an algebraic expression. This is a common practice in algebra, where we use letters to represent unknown values. The phrase given was "Twice a number, decreased by 72."

Algebra shows us the power of transforming words into mathematical symbols, like using the letter \(x\) to denote an "unknown number." Through understanding and using algebra, we can decipher and solve a wide array of real-world problems.

Variables

Variables are essential components of algebra as they represent unknown or changeable values. In our example, the unknown number was represented by the variable \(x\). Instead of just guessing the value, we use \(x\) as a substitute or a placeholder until we have more information about it.

Think of variables like empty boxes that can hold any number. They allow you to write equations and expressions that work for numerous cases. For instance, in the expression \(2x - 72\), \(x\) is that empty box. By figuring out what \(x\) stands for, we can solve the whole equation.

Using variables, like \(x\), makes it easier for us to understand problems where the exact numbers aren't clear or when they can change. This flexibility is why variables are crucial in algebra, allowing us to express complex relationships clearly and simply.

Basic Operations

In algebra, basic operations like addition, subtraction, multiplication, and division help us manipulate expressions and equations. To build the algebraic expression for the given phrase, we needed to combine these basic operations.

Let's break down the steps:

• Multiplication: "Twice a number" means multiplying the number (\(x\)) by 2, resulting in \(2x\).
• Subtraction: "Decreased by 72" means we subtract 72 from the product \(2x\), giving us the complete expression \(2x - 72\).

Using these operations allows us to construct expressions that are not only compact but also flexible. Understanding how and when to use each operation is vital for solving different algebraic problems. Hence, mastering these operations is fundamental to progressing in algebra.