Problem 103 - Intermediate Algebra

Question

When asked to write an algebraic expression for " 8 more than a number," you wrote \(x+8\) and another student wrote \(8+x\). Are both expressions correct? Explain your answer.

Step-by-Step Solution

Answer

Yes, both expressions \(x + 8\) and \(8 + x\) are correct due to the commutative property of addition.

Step 1: Understand the Expression

The phrase "8 more than a number" indicates that we are adding 8 to some unknown number, which we will denote with a variable, such as \(x\).

Step 2: Formulate the Algebraic Expression

To translate the phrase "8 more than a number" into algebra, the number (\(x\)) is identified first, and then 8 is added. This can be written as the algebraic expression \(x + 8\).

Step 3: Commutative Property of Addition

According to the commutative property of addition, the order in which we add numbers does not change the result. Therefore, \(x + 8\) is equivalent to \(8 + x\). Both expressions mean adding 8 to the unknown number, \(x\).

Step 4: Verify Both Expressions

Since the commutative property applies, both \(x + 8\) and \(8 + x\) are valid algebraic expressions for "8 more than a number." Both represent the process of adding 8 to the variable \(x\).

Key Concepts

Commutative Property

The commutative property is a fundamental concept in algebra that helps simplify expressions and equations. It states that the order of addition or multiplication does not affect the final result. For addition, this can be expressed as:

• \(a + b = b + a\)

This means if you have two numbers or terms and you add them, you can swap their order without changing the sum. In the context of the problem, both expressions \(x + 8\) and \(8 + x\) are equivalent.
It shows that the order of adding 8 to the variable \(x\) does not matter. This property is particularly helpful when rearranging terms in larger expressions, making computations more straightforward and manageable.

Unknown Variables

When working with algebraic expressions, we often encounter unknowns or variables. A variable is a symbol, usually a letter, used to represent an unknown value in an expression or equation. In our exercise, the unknown variable is \(x\).
This variable represents the number that we are adding 8 to. Using a variable is a powerful concept because it allows us to create general formulas and solve problems where the specific numbers might change.
Eventually, understanding variables lays the groundwork for solving equations, where you determine the value of the variable that makes the equation true. Variables offer flexibility and are essential to the algebraic language.

Formulating Expressions

Creating algebraic expressions from word problems is a key skill in algebra. It requires translating language into mathematical operations using numbers and variables. In our case, "8 more than a number" translates to:

• "A number" indicates a variable, often represented as \(x\).
• "More than" suggests addition.
• "8" is the constant being added.

So, the phrase becomes \(x + 8\), and thanks to the commutative property, \(8 + x\) is equally valid.
This formulation allows us to write functions and equations that describe relationships and solve problems effectively.