Problem 8 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of addition to write an equivalent algebraic expression. $$x+7$$

Step-by-Step Solution

Answer

The equivalent algebraic expression to \(x + 7\) using the commutative property of addition is \(7 + x\).

Step 1: Identify the Properties of Addition

The commutative property of addition is one of the fundamental properties of mathematics. It states that the order of the numbers does not affect the sum, that means, if you are adding two numbers, you can swap their order and still get the same answer. Now, this property is going to be applied to the given algebraic expression \(x + 7\).

Step 2: Apply the Commutative Property of Addition

Applying the commutative property of addition to the expression \(x + 7\), it can be reordered as \(7 + x\). Therefore, \(7 + x\) is an equivalent algebraic expression to \(x + 7\).

Key Concepts

Properties of Addition

Understanding the properties of addition is fundamental for mastering arithmetic and algebra. One of the central properties is the Commutative Property of Addition. This property tells us that changing the order of the numbers we are adding does not change their sum.

For instance, if you have two numbers, let's say 3 and 5, you can add them in any order:

• 3 + 5 = 8
• 5 + 3 = 8

As you see, the sum remains the same regardless of the order. This concept helps a lot, especially in algebra, where variables are present. It allows you to rearrange terms in expressions, making them easier to work with. In the exercise expression \(x + 7\), you can apply the commutative property to rewrite it as \(7 + x\). This flexibility can be handy in more complex calculations.

Equivalent Algebraic Expressions

Equivalent algebraic expressions are expressions that have the same value for all values of their variables. When we use the commutative property of addition, like with \(x+7\) and \(7+x\), we create such equivalent expressions.

Understanding equivalence is essential because it enables algebraic manipulation and simplification. You can change expressions into more convenient forms without altering their fundamental meaning or value.
To test if two expressions are equivalent, you can substitute the variable with a number to check if they both produce the same outcome.

• For example, with \(x = 2\), both \(x + 7 = 2 + 7 = 9\) and \(7 + x = 7 + 2 = 9\).

This confirms that both expressions are indeed equivalent. Having the ability to recognize and create equivalent expressions increases your problem-solving skills in algebra.

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the correct sequence when solving expressions. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this set of rules ensures that calculations are carried out in the correct order.

While the commutative property allows you to rearrange terms, the order of operations guides you on how to structure those computations, especially when multiple operations are involved. For the expression involving added terms such as \(x + 7\), the primary focus is on the addition part.
Since addition is at the bottom of the hierarchy in order of operations, if there are no parentheses indicating a higher priority, performing addition is straightforward. The commutative property lets you rearrange these terms as \(7 + x\), which doesn't change the operation order needed. Understanding these rules prevents errors and equips you with the proper tools to tackle complex numerical and algebraic expressions confidently.