Problem 20 - Beginning and Intermediate Algebra

Question

Use a commutative or an associative property to complete each statement. State which property is used. \((-2+3)+6=-2+\)( ___ +6)

Step-by-Step Solution

Answer

Associative Property of Addition: \(-2+(3+6)\).

Step 1: Identify Original Statement

The given statement is \((-2+3)+6=-2+(\text{___}+6)\). The goal is to find the missing number and state which property was used.

Step 2: Rewrite Using the Associative Property

Use the associative property of addition to rewrite the expression. The associative property states that the grouping of numbers does not affect their sum. For example, \((a+b)+c = a+(b+c)\).

Step 3: Apply Associative Property

Apply the associative property to the given statement: \((-2+3)+6 = -2+(3+6)\).

Step 4: Complete the Statement

Fill in the missing number: \(-2+(3+6)\). So, the statement is \((-2+3)+6 = -2+(3+6)\).

Step 5: State the Property

The property used to complete the statement is the associative property of addition.

Key Concepts

Commutative Property

The commutative property is a foundational concept in mathematics. It tells us that the order in which we add or multiply numbers does not change the result. For addition, the commutative property is written as: \(a + b = b + a\).
This property is immensely useful in simplifying and solving equations because it allows us to rearrange numbers for easier computation.
For example, \2 + 3 = 3 + 2\. Whether you add 2 plus 3 or 3 plus 2, the result is always 5. In multiplication, \(a \times b = b \times a\). This means \4 \times 7 = 7 \times 4\. Both ways give us 28.