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Chapter 1: Problem 1

Use a commutative property to complete each statement. See Examples 1 and 3. $$ x+16= $$

Short Answer

Expert verified
16 + x

Step by step solution

01

Understanding the Commutative Property

The commutative property states that changing the order of numbers in an addition or multiplication operation does not change the result. For addition, this property is expressed as \( a + b = b + a \). This means we can swap the positions of the numbers being added or the factors being multiplied.
02

Apply the Commutative Property of Addition

Use the commutative property of addition to rewrite the expression \( x + 16 \). According to this property, you can switch the two terms, so it becomes \( 16 + x \).
03

Verify the Application

Check that both expressions \( x + 16 \) and \( 16 + x \) are equivalent by adding 16 to a variable \( x \). Since addition is commutative, the order of terms does not affect the sum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition - Understanding the Basics
Addition is a fundamental arithmetic operation, often the first one we learn in mathematics. It involves combining two or more numbers to find their total. The numbers that you add together are called "addends," and the result is known as the "sum."
For example:
  • In the expression \(3 + 5\), 3 and 5 are addends, and their sum is 8.
Addition is all about combining quantities, and it's crucial in everyday life. Whether you're adding up items in a shopping cart or summing points in a game, addition is at play. It's straightforward and doesn't change based on the way we group or order the numbers, thanks to properties like the commutative property.
Algebra and the Role of Variables
Algebra can be thought of as a bridge between arithmetic and more complex mathematics. It introduces the use of variables鈥攕ymbols, often letters, that stand in for numbers in equations.
For example:
  • In the equation \(x + 16\), \(x\) is a variable that represents a number we may not yet know.
By using variables, algebra allows us to express general rules and patterns that apply in many situations. This can be incredibly powerful, as it helps us solve real-world problems using mathematical models. The introduction of variables transforms simple arithmetic into more abstract thought processes, letting students explore patterns and relationships in numbers and operations.
Equivalent Expressions Explained
Equivalent expressions are expressions that, although they may look different, represent the same value for all possible values of the variables contained within them. Recognizing equivalent expressions is crucial in algebra because it helps simplify problems and solve equations.
Consider the expressions:
  • \(x + 16\) and \(16 + x\).
Using the commutative property of addition鈥攚hich tells us we can switch the order of addends鈥攖hese expressions are equivalent. This means no matter what number \(x\) is, both expressions give the same sum.Understanding and identifying equivalent expressions can make complex problems more approachable and ensure you鈥檙e on the right path when working on algebraic operations.

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Most popular questions from this chapter

Translate each phrase to an algebraic expression. Use " \(x\) " to represent "a number." The difference of -3 and a number.

Subtract. \(12-(-5)\)

Subtract. \(-6-5\)

If \(p\) is a positive number and \(n\) is a negative number, determine whether each statement is true or false. Explain your answer. $$ |n-p| \text { is always a positive number. } $$

Translate each phrase to an algebraic expression. Use " \(x\) " to represent "a number." Add a number and -36 .
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