Problem 117 - Prealgebra

Question

Match each statement on the left with the property that justifies it on the right. a. Distributive property b. Associative property c. Commutative property d. Commutative and associative properties $$(x+5)+1=1+(x+5)$$

Step-by-Step Solution

Answer

Commutative property.

Step 1: Identify the Switch

Examine the expression \((x+5)+1=1+(x+5)\). Notice that the terms are rearranged from \((x+5)+1\) to \(1+(x+5)\), indicating a change in the order of addition.

Step 2: Recognize the Property

Recall that the Commutative Property allows the reordering of numbers in addition or multiplication, so \(a+b\) becomes \(b+a\). In this expression, changing \((x+5) + 1\) to \(1 + (x+5)\) fits the definition of the Commutative Property.

Key Concepts

Commutative Property

The commutative property is a fundamental principle in mathematics. It refers to the ability to rearrange elements in addition or multiplication without changing the outcome. In simple terms, the order in which you add or multiply numbers doesn't matter—they always give the same result.

• For addition: If you have two numbers, say, 3 + 4, it's the same as 4 + 3. Thus, 3 + 4 = 4 + 3 = 7.
• For multiplication: 6 × 2 gives the same result as 2 × 6, which is 12.

Commutative property applies to real numbers, integers, rational numbers, and at times to matrices and vectors under addition. It's important to remember that subtraction and division do not follow this property—changing their order affects the result.

Associative Property

The associative property involves grouping. It states that how you group numbers in addition or multiplication doesn’t change their sum or product.
This property is vital because it allows flexibility in calculations, especially complex ones involving many numbers.

• For addition: If you have a set of numbers, say \((2 + 3) + 4 = 2 + (3 + 4)\). Both groupings result in 9.
• For multiplication: Consider \((5  2)  3\) and \(5  (2  3)\). Each setup gives you 30.

The associative property highlights operations' flexibility, making it easier to simplify and solve equations.

Distributive Property

The distributive property is a powerful tool in mathematics, especially when dealing with expressions that need simplification. It combines addition and multiplication, showing how a term multiplied by a sum equals the sum of individual products.
Consider the expression, \(a(b + c)\). Using the distributive property, you multiply \(a\) with each term inside the parenthesis, giving \(a  b + a  c\).
Here's a practical example:

• Let’s say you have \(3(4 + 5)\). Applying the distributive property gives \(3  4 + 3  5\) which simplifies to 12 + 15, resulting in 27.

It’s particularly useful in algebra to expand expressions and solve equations by breaking them into manageable parts before recombining.