Problem 22 - Algebra 2

Question

Write an algebraic expression to represent each verbal expression. twice the sum of a number and 8

Step-by-Step Solution

Answer

The algebraic expression is \( 2(x + 8) \).

Step 1: Understand the Verbal Expression

The phrase "twice the sum of a number and 8" can be broken down into two parts: the "sum of a number and 8" and then "twice" this sum. Let's denote the unknown number as \( x \).

Step 2: Express the Sum

First, consider the phrase "the sum of a number and 8". This means we are adding a number (which we've called \( x \)) to 8. This can be written algebraically as: \( x + 8 \).

Step 3: Apply the Concept of 'Twice'

The term "twice" means to multiply by 2. So, when the expression says "twice the sum," it means we need to multiply the previous expression (\( x + 8 \)) by 2: \( 2(x + 8) \).

Step 4: Write the Complete Algebraic Expression

Combine the information from the previous steps to form the complete expression: The algebraic expression for "twice the sum of a number and 8" is \( 2(x + 8) \).

Key Concepts

Verbal Expressions

When dealing with algebra, verbal expressions are phrases we use to describe mathematical operations without using numbers or symbols. They serve as a bridge between the language we speak and the language of math. In a classroom setting, you might hear your teacher ask you to "translate" these verbal expressions into algebraic ones. This process involves identifying keywords and knowing what they mean mathematically.

For example, when you see or hear "twice the sum of a number and 8," each part plays a role in building our corresponding algebraic expression. Here, the phrase breaks down into two components: "twice" and "the sum of a number and 8."

• "Twice" indicates a multiplication operation.
• "The sum of a number and 8" implies an addition operation with a variable and a constant.

Recognizing these keywords can help you create meaningful algebraic expressions efficiently.

Sum

In mathematics, the term "sum" refers to the result of adding two or more quantities together. In the context of our exercise, "the sum of a number and 8" means that we are adding together an unknown number and the number 8.

To write this as an algebraic expression, we use a variable (let's say \( x \) for our unknown number) and simply add 8 to it. The expression becomes \( x + 8 \). This expression tells us that whatever value \( x \) takes, we should add 8 to it to find the outcome.

Understanding what a "sum" embodies helps you decode the phrasing in verbal questions and convert them effectively into algebraic language.

Multiplication

Multiplication is one of the fundamental operations in algebra. It is denoted by various symbols like \( \times \), \( * \), or simply by juxtaposition (placing numbers or variables side-by-side). In our exercise, the word "twice" signals a multiplication operation.

"Twice" literally implies "two times". Thus, when we see "twice," we interpret it as "multiply by 2." For our expression \( x + 8 \), applying "twice" means multiplying the entire sum by 2. This operation is represented as \( 2(x + 8) \).

This multiplication ensures that whatever the result of \( x + 8 \) is, it gets doubled, maintaining the relationship as described in the original verbal expression.

Variables

Variables are symbols used to represent unknown or changeable numbers in algebraic expressions and equations. They act as placeholders for numbers we either don't know yet or that can change depending on the context.

In our exercise, we use the variable \( x \) to stand in for "a number," which is unspecified in the verbal problem "twice the sum of a number and 8." Using a variable allows us to write a general expression, \( 2(x + 8) \), that can work for any value of \( x \).

This flexibility is one of algebra's core strengths, enabling us to solve equations and perform calculations that apply universally rather than just to one specific scenario.