Problem 71 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression. Use \(x\) for the unknown number. The product of 9 and the sum of a number and 20

Step-by-Step Solution

Answer

\(9(x + 20)\)

Step 1: Identify the Phrase Components

The phrase is "The product of 9 and the sum of a number and 20." This can be broken down into two main parts: 1. "The sum of a number and 20." 2. "The product of 9 and (the part from point 1)."

Step 2: Express the Sum as an Expression

For the first part, "the sum of a number and 20," use \(x\) to represent the unknown number. Thus, the expression becomes \(x + 20\).

Step 3: Express the Product as an Expression

Now use the second part, "the product of 9 and (the sum)," which means you multiply 9 by the first part's result. So, the entire expression is \(9 \times (x + 20)\).

Step 4: Simplify the Expression

While the problem is solved with \(9 \times (x + 20)\), this can also be expressed in a simpler form without changing its meaning: \(9(x + 20)\).

Key Concepts

Product of Numbers

Understanding the concept of the product of numbers is an essential component of solving algebraic phrases and expressions. In simple terms, when you see the word "product," it means multiplication. When working with algebraic expressions, it's quite common to find instructions like "the product of two numbers" or "the product of a number and a quantity."

In our given exercise, "the product of 9 and the sum of a number and 20" asks us to multiply 9 by another expression. Here, the numerical value 9 and the expression within the parentheses are multiplied together, creating the product. When you multiply a number by an algebraic expression, you distribute that number across every term inside the parentheses in the expression.

• The multiplication can be represented by the symbol \( \times \) or just placing the number next to the parentheses, like in \( 9(x + 20) \).
• The product remains the same regardless of how you write it: \( 9 \times (x + 20) \) or \( 9(x + 20) \).

Sum of a Number

The term "sum" in algebra often refers to the result of adding two or more numbers or expressions. When you need to express a "sum," it means you'll be using addition. For instance, the sum of a number and another value involves combining them with a plus sign.

In the exercise, "the sum of a number and 20" translates to the algebraic expression \( x + 20 \). Here, \( x \) is our variable representing an unknown number, and we're adding it to 20.

• To form a sum, use the plus sign, \(+\).
• Write the unknown variable and the constant (like 20) next to each other separated by the plus sign, as seen in \( x + 20 \).
• Always consider the order; \( x + 20 \) means you start with \( x \) and add 20 to it.

This basic concept of sum is fundamental in writing and understanding algebraic expressions.

Unknown Variable

An unknown variable in algebra is commonly represented by letters, like \( x \), \( y \), or \( z \). This variable is a placeholder for a number that we do not yet know. Understanding how to work with these variables is crucial for solving algebraic equations and expressions.

In our phrase, the "unknown number" is represented by \( x \), making it easier to set up our algebraic equation. Whenever you encounter an unknown variable, keep in mind:

• It's used to represent a quantity that needs to be found or is unknown.
• You can manipulate these variables just like numbers, using operations such as addition, subtraction, multiplication, or division.
• Variables allow for generalization and solving for different instances of a problem.

Recognizing \( x \) as the unknown variable helps structure the formula and move towards finding its solution step-by-step. Recognizing this concept will simplify understanding various algebraic problems.