Problem 42 - 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. $$ 6(5 t)-6(7 r)=6(5 t-7 r) $$

Step-by-Step Solution

Answer

Distributive property.

Step 1: Identify the comparable expressions

Look at the two expressions given in the problem: First Expression: \[ 6(5t) - 6(7r) \] Second Expression: \[ 6(5t - 7r) \]

Step 2: Apply Distributive Property

The distributive property states: \[ a(b + c) = ab + ac \] In this case, compare it against \[ 6(5t) - 6(7r) = 6(5t - 7r) \] Identify that on the right hand side, we use the distributive property to combine terms inside the parenthesis.

Step 3: Conclude

Given the equivalence, and how the property is applied, conclude that this is an example of the distributive property, where the constant term (6) is distributed across the terms inside the parenthesis.

Key Concepts

Commutative Property

The commutative property is one of the fundamental properties of arithmetic operations. It states that the order in which you add or multiply numbers does not change the result. For addition, it can be expressed as: \( a + b = b + a \). Similarly, for multiplication, it states: \( a \times b = b \times a \).

This property is called 'commutative' because the numbers can 'commute' or move around each other.

• Example for addition: \( 4 + 5 = 5 + 4 \)
• Example for multiplication: \( 3 \times 7 = 7 \times 3 \)

It's crucial to note that this property does not apply to subtraction or division.

Associative Property

The associative property involves grouping. It states that the way in which numbers are grouped in addition or multiplication does not affect the sum or product. For addition, the associative property is: \( (a + b) + c = a + (b + c) \). For multiplication, it can be written as: \( (a \times b) \times c = a \times (b \times c) \).

Here, the numbers are associated or grouped in different ways but the results remain the same.

• Example for addition: \( (2 + 3) + 4 = 2 + (3 + 4) \)
• Example for multiplication: \( (2 \times 3) \times 4 = 2 \times (3 \times 4) \)

Remember, like the commutative property, the associative property does not work with subtraction or division.

Identity Property

The identity property consists of addition and multiplication and defines special elements called 'identities'. For addition, the identity property states: \( a + 0 = a \). For multiplication, it is: \( a \times 1 = a \).

The identity for addition is 0 because adding 0 to a number leaves it unchanged.

• Example for addition: \( 7 + 0 = 7 \)
• Example for multiplication: \( 9 \times 1 = 9 \)

Similarly, the identity for multiplication is 1 as multiplying a number by 1 does not change the number. Understanding identity properties will help in simplifying algebraic expressions and solving equations efficiently.

Inverse Property

The inverse property involves the concept of opposites. For addition, the inverse property is about finding the additive inverse or the negative of a number, such that adding a number and its inverse gives 0: \( a + (-a) = 0 \). For multiplication, it focuses on the multiplicative inverse (reciprocal), where multiplying a number by its reciprocal results in 1: \( a \times \frac{1}{a} = 1 \) (where \(a eq 0\)).

Inverse properties are particularly useful when solving equations as they allow you to 'cancel out' values.

• Example for addition: \( 8 + (-8) = 0 \)
• Example for multiplication: \( 5 \times \frac{1}{5} = 1 \)

Recognizing and utilizing the inverse properties can simplify your calculations remarkably.