Problem 15 - Beginning and Intermediate Algebra

Question

Use a commutative or an associative property to complete each statement. State which property is used. $$ -15+9=9+ _____ $$

Step-by-Step Solution

Answer

-15

Step 1: Identify the problem

Given is the expression \(-15 + 9 = 9 + \text{\rule{1cm}{0.4pt}}\). The task requires applying a property to determine the missing value.

Step 2: Understand the commutative property

The commutative property states that for all real numbers a and b, the sum remains the same regardless of their order: \(a + b = b + a\).

Step 3: Apply the commutative property

Rewrite the original expression using the commutative property: \(-15 + 9 = 9 - 15\).

Step 4: Complete the statement

Fill in the missing value in the expression to obtain: \(-15 + 9 = 9 - 15\).

Step 5: State the property used

The commutative property of addition is used to solve this problem.

Key Concepts

commutative property

The commutative property is a fundamental principle in algebra. It allows us to move the numbers around in an equation without changing the result. For addition, this property states: if you have two numbers, say a and b, then switching their order does not affect their sum. Mathematically, this can be written as: \(a + b = b + a\). For example, if we look at the problem \(-15 + 9 = 9 + \text{_____}\) from the exercise, we can see that switching the order of the numbers still keeps the sum the same. Hence, \(-15 + 9 = 9 + (-15)\). This simple yet powerful property is especially useful in simplifying algebraic expressions and solving equations. Remembering the commutative property can help you quickly rearrange numbers and see solutions more clearly.

addition in algebra

Addition in algebra follows the same basic principles as addition in arithmetic, but with more abstract elements like variables. When you add two numbers or algebraic expressions, you're essentially combining their values. For example, adding \(a\) and \(b\) will give you \(a + b\), regardless of whether \(a\) and \(b\) are numbers, variables, or a combination of both.
It's important to apply algebraic properties, such as the commutative and the associative properties, to simplify expressions or solve equations.
In the given exercise, understanding how addition works allows us to utilize the commutative property to rewrite \(-15 + 9\) as \(9 + (-15)\).
Being comfortable with addition in algebra will set a solid foundation for tackling more complex mathematical problems.

associative property

The associative property is another key concept in algebra. It tells us that when we're adding three or more numbers, the way in which they are grouped does not change the sum. Mathematically, it is expressed as: \((a + b) + c = a + (b + c)\).
For example, if we have three numbers \(1, 2,\) and \(3\), we can add them in different groupings: \((1 + 2) + 3\) or \(1 + (2 + 3)\). Both groupings will result in the same sum, \(6\).
This property is particularly useful when dealing with complex algebraic expressions, as it allows for flexibility in calculations. While the associative property is different from the commutative property, both work together to simplify expressions. Although the associative property was not directly used in the given exercise, understanding it complements our knowledge of how to manipulate and simplify algebraic expressions effectively.