Problem 38 - 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. $$ 9(t+s)=9 t+9 s $$

Step-by-Step Solution

Answer

Distributive property

Step 1: Recognize the form of the equation

Examine the given equation $$ 9(t+s)=9t+9s $$. Notice that a single number (in this case, 9) is multiplied by a sum within parentheses, and then that number is multiplied by each term inside the parentheses separately.

Step 2: Identify the property

The form matches the distributive property, which states that for all numbers a, b, and c, $$ a(b+c) = ab + ac $$. Here, $$ 9(t+s) $$ is distributed across the sum, resulting in $$ 9t + 9s $$.

Key Concepts

commutative property

The commutative property is one of the fundamental properties of numbers.
It states that for any two numbers, the order in which you add or multiply them does not change the result.

For example, if you have the numbers 3 and 5, then:

• In addition: \(3 + 5 = 5 + 3 = 8\)
• In multiplication: \(3 times 5 = 5 times 3 = 15\)

It's important to note that this property is only valid for addition and multiplication. It does not apply to subtraction or division.
For instance, \(5 - 3 ≠ 3 - 5\) and \(6 div 2 ≠ 2 div 6\).
Understanding the commutative property helps in simplifying calculations and recognizing patterns in mathematics.

associative property

The associative property is another vital property of numbers.
It states that the way numbers are grouped in addition or multiplication does not change their result.

For any three numbers (a, b, and c):

• In addition: \((a + b) + c = a + (b + c)\)
• In multiplication: \((a times b) times c = a times (b times c)\)

Here's an example to clarify:

If you have 2, 3, and 4:

• For addition: \(2 + (3 + 4) = (2 + 3) + 4 = 9\)
• For multiplication: \(2 times (3 times 4) = (2 times 3) times 4 = 24\)

These groupings do not matter, the result will always be the same.
This property is useful when dealing with complex calculations, as it allows us to re-group numbers in ways that make the calculations easier.

identity property

The identity property involves the numbers that do not change the value of other numbers in an operation.
There are two types of identity properties:

• Additive Identity: This involves the number 0. Any number added to 0 remains unchanged. For example, \(7 + 0 = 7\).
• Multiplicative Identity: This involves the number 1. Any number multiplied by 1 remains the same. For instance, \(9 times 1 = 9\).

These properties confirm that there exist certain 'special' numbers that, when applied in addition or multiplication, leave other numbers unaffected.
This helps to maintain the consistency of results in various calculations and provides a foundation for more complex mathematical operations.

inverse property

The inverse property focuses on 'undoing' effects of operations.
There are two main types of inverse properties:

• Additive Inverse: For any number a, there exists a number -a, such that \(a + (-a) = 0\). For example, if you have 8, its additive inverse is -8, since \(8 + (-8) = 0\).
• Multiplicative Inverse: For any number a (except 0), there exists a number \(\frac{1}{a}\) (also called its reciprocal), such that \(a times \frac{1}{a} = 1\). For instance, if the number is 4, its reciprocal is \(\frac{1}{4}\), and \(4 times \frac{1}{4} = 1\).

These properties are crucial as they allow solutions and simplification in equations.
By understanding how to 'cancel out' terms using their inverses, students can solve more complex equations with confidence.