Problem 115 - 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 $$x+5=5+x$$

Step-by-Step Solution

Answer

The equation is justified by the Commutative Property of Addition.

Step 1: Identify Property

Examine the equation given: \(x + 5 = 5 + x\). Notice that the order of the terms on either side of the equation has been swapped.

Step 2: Apply the Commutative Property

The Commutative Property of Addition states that the order of two numbers being added does not change their sum: \(a + b = b + a\). This is exactly what has been done in the equation. Therefore, the equation is justified by the Commutative Property.

Key Concepts

Commutative Property

In the world of math, the Commutative Property is a fundamental principle that simplifies expressions and calculations. The Commutative Property of Addition dictates that the order in which you add numbers does not affect the sum. For instance, if you have two numbers, like 4 and 7, adding them in any order gives you the same result:

• 4 + 7 = 11
• 7 + 4 = 11

This property also applies to multiplication. Whether it’s 3 \( \times \) 5 or 5 \( \times \) 3, the answer remains 15. However, it's important to remember that the Commutative Property does not extend to subtraction or division. Swapping the order in subtraction or division gives you different results, so it’s essential to keep the original order in mind when dealing with these operations. Try experimenting with different numbers and operations to see this property in action!

Associative Property

The Associative Property is another powerful tool to facilitate simplification, particularly when dealing with multiple numbers. This property is concerned with grouping. For addition and multiplication, the way numbers are grouped doesn’t impact the sum or product.Consider this grouping of three numbers: (2 + 3) + 4. With the Associative Property, you can regroup them as 2 + (3 + 4), and the result will still be the same. Let's break it down further:

• (2 + 3) + 4 = 5 + 4 = 9
• 2 + (3 + 4) = 2 + 7 = 9

Similarly, for multiplication, let's say (2 \( \times \) 3) \( \times \) 4. Reordering the multiplication does not change the result:

• (2 \( \times \) 3) \( \times \) 4 = 6 \( \times \) 4 = 24
• 2 \( \times \) (3 \( \times \) 4) = 2 \( \times \) 12 = 24

This property, much like the Commutative Property, does not apply to subtraction or division, which means how numbers are grouped in these operations will alter the results.

Distributive Property

The Distributive Property is an invaluable rule in algebra that combines both multiplication and addition. It allows you to multiply a sum by a number by multiplying each addend separately and then adding the products. Algebraically, it looks like this: \[ a \times (b + c) = a \times b + a \times c \]Imagine you have 3 \( \times \) (4 + 5). You can distribute the 3 and simplify in the following way:

• 3 \( \times \) 4 + 3 \( \times \) 5 = 12 + 15 = 27

Alternatively, you could first add 4 and 5, then multiply:

• 3 \( \times \) 9 = 27

The result is the same in either case. Beyond simplifying calculations, the Distributive Property is pivotal when expanding algebraic expressions. For example, the expression 2(x + 3) will expand to 2x + 6, applying this property. Mastering the Distributive Property is essential for solving equations efficiently and enhancing your algebraic skills.