Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of addition to write an equivalent algebraic expression. $$5(x+3)$$

Step-by-Step Solution

Answer

The equivalent algebraic expression using the commutative property of addition for the expression \(5(x+3)\) is \(15 + 5x\).

Step 1: Distributive Property

First, apply the distributive property of multiplication over addition. This property states that \(a(b + c) = ab + ac\). So, it will be \(5x + 15\).

Step 2: Apply Commutative Property

Now apply the commutative property of addition. This property states that \(a + b = b + a\). Using this property, the expression can be rewritten as \(15 + 5x\).

Key Concepts

Distributive Property of Multiplication

The distributive property of multiplication over addition is a fundamental principle in algebra. It allows us to simplify expressions and solve equations more easily. According to this property, when you multiply a sum by a number, you can multiply each addend of the sum by the number and then add the products. This is formally expressed as \( a(b + c) = ab + ac \).

Let's understand by using the provided exercise where we have \(5(x + 3)\). We distributed the \(5\) across the terms inside the parentheses: \(5 \times x + 5 \times 3\) giving us \(5x + 15\). This is crucial because it allows us to transform a multiplication problem into an addition problem, making it simpler to work with, especially if the terms inside the parentheses are more complex or involve variables.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the expression \(5(x+3)\), \(5\) is a coefficient, \(x\) is a variable, and \(3\) is a constant. When working with algebraic expressions, the goal is often to simplify or manipulate them to get an equivalent expression using properties like the distributive property.

An understanding of how to work with algebraic expressions is integral to solving mathematical problems. It allows you to represent complicated relationships through an equation or inequality that can be analyzed and simplified to discover unknown values.

Equivalent Expressions

Equivalent expressions are expressions that, although they may look different, represent the same quantity. The exercise shows how you can use different properties of arithmetic, like the distributive property and the commutative property, to write equivalent expressions. From \(5(x+3)\) we used the distributive property to get \(5x + 15\), and then the commutative property to get \(15 + 5x\). These expressions are equivalent because they both yield the same result when the variable \(x\) is assigned a value.

Recognizing equivalent expressions is a valuable skill. It allows for many problem-solving techniques and aids in understanding the relationships between different parts of an algebraic equation.