Problem 23 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ 4+15=15+4 $$

Step-by-Step Solution

Answer

Commutative Property

Step 1: Identify the Property

First, recognize the structure of the given equation. The equation is 4 + 15 = 15 + 4.

Step 2: Understand the Property

The commutative property states that the order in which two numbers are added does not change the sum. In mathematical terms, for any numbers a and b, a + b = b + a.

Step 3: Compare the Equations

Compare the given equation with the commutative property definition: 4 + 15 indeed equals 15 + 4. This shows that the order of addition does not affect the result.

Step 4: Determine the Property

Since switching the order of the numbers in the addition does not change the result, this example is a commutative property.

Key Concepts

commutative property

The commutative property is a basic but very important concept in mathematics. It tells us that the order in which we add or multiply numbers does not affect the result. If you see an equation like 4 + 15 = 15 + 4, you are looking at the commutative property in action.
This property can be written in general terms as follows:
dot a + b = b + a< ordot ab = ba. This means that whether you add 4 to 15 or 15 to 4, the sum will remain 19. Similarly, with multiplication, for any numbers a and b, a x b will always be equal to b x a. Understanding this property helps simplify and solve mathematical equations more easily.

algebraic properties

Algebraic properties are rules that apply to numbers and variables in algebra. They help us understand and manipulate equations. The main algebraic properties include:

• Commutative Property
• Associative Property
• Distributive Property
• Identity Property
• Inverse Property

Each of these properties has its own unique rules and applications.
Recognizing which property is being used in a problem can make solving it much easier and quicker. For example, in the equation 4 + 15 = 15 + 4, identifying the use of the commutative property helps us understand that the sum stays the same regardless of the order.

addition properties

The properties of addition are fundamental in math. Knowing them helps us solve problems more efficiently. Here are the key addition properties:

• Commutative Property: a + b = b + a
• Associative Property: (a + b) + c = a + (b + c)
• Identity Property: a + 0 = a
• Inverse Property: a + (-a) = 0

These properties ensure that addition is consistent and predictable.
Wherever you see numbers being added, understanding these properties helps you know how you can group or order the numbers without changing the final sum. In our example, 4 + 15 = 15 + 4, the commutative property is used.

mathematical equations

Mathematical equations are statements that show the equality of two expressions. They often include numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.
Every equation follows certain algebraic properties that help us understand and solve them. For example, in the equation 4 + 15 = 15 + 4, we can recognize the commutative property of addition at work.
Understanding these principles not only helps in solving equations correctly but also in understanding the structure of more complex mathematical expressions. Equations are fundamental in math because they describe relationships between quantities and allow us to make calculations and predictions.