Problem 23 - Intermediate Algebra

Question

Complete each statement so that the indicated property is illustrated. See Example 1. $3+7= ______ Commutative property of addition

Step-by-Step Solution

Answer

3 + 7 = 7 + 3 using the commutative property of addition.

Step 1: Identify the Commutative Property

The commutative property of addition states that changing the order of the numbers in a sum does not change the result. For any two numbers, \( a \) and \( b \), \( a + b = b + a \).

Step 2: Apply to Given Expression

The original expression is \( 3 + 7 \). To illustrate the commutative property, we can write this as \( 7 + 3 \) and it will still equal the same value.

Step 3: Complete the Statement

Substituting the expression using the commutative property, the complete statement becomes \( 3 + 7 = 7 + 3 \).

Key Concepts

Properties of Addition

Addition is a fundamental mathematical operation, and there are several properties associated with it to make it easier to understand and work with.
One of the main properties is known as the **Commutative Property of Addition**, which means that no matter how two numbers are ordered, their sum remains the same.
This is mathematically expressed as:

• Given any two numbers, \(a\) and \(b\), the equation \(a + b = b + a\) holds true.
• For example, with the numbers 3 and 7, both \(3 + 7 = 10\) and \(7 + 3 = 10\).

Another property is the **Associative Property**, which involves three or more numbers. It states that how you group the numbers does not affect the sum.
Lastly, there's the **Identity Property of Addition**, which involves zero. The sum of any number and zero is the number itself:

• For any number \(a\), \(a + 0 = a\).

Order of Operations

The order of operations is a set of rules that dictates the sequence in which operations should be performed to accurately evaluate mathematical expressions.
Without these rules, solving expressions can lead to different answers and confusion.
The order of operations can be remembered by the acronym **PEMDAS**:

• **P**arentheses (solve expressions inside parentheses first).
• **E**xponents (next, solve exponents or powers).
• **M**ultiplication and **D**ivision (from left to right).
• **A**ddition and **S**ubtraction (from left to right).

When performing calculations, follow these rules carefully to arrive at the correct answer.
This order ensures that everyone arrives at the same result when calculating the same expression.
For example, in the expression \(2 + 3 \times 4\), according to PEMDAS, you first multiply 3 and 4, then add 2, resulting in the correct answer of 14.

Mathematical Expressions

Mathematical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division.
They can be simplified or solved based on the rules established by properties of numbers and operations.
An important aspect of working with expressions is understanding how to **interpret and simplify** them accurately:

• Using properties of operations, like the commutative property, can allow rearranging terms in a way that might be simpler to calculate or understand.
• Recognizing which operations to perform first is crucial; this knowledge comes with understanding order of operations.

Expressions do not always have an equal sign; however, once simplified or evaluated, they often form equations.
By practicing the evaluation of expressions, students can develop a stronger foundation in algebra and arithmetic, ensuring they can tackle more complex calculations in the future.