Problem 73 - Beginning and Intermediate Algebra

Question

Name the properties illustrated by each true statement. See Example 6 \(2+(x+5)=(2+x)+5\)

Step-by-Step Solution

Answer

Associative Property of Addition.

Step 1: Identify the Operation

First, we need to identify the mathematical operation used in the equation. The operations in this statement are addition.

Step 2: Recognize the Associative Property

In the expression \(2+(x+5)=(2+x)+5\), the grouping of numbers is changing, but the numbers themselves and the operation (addition) remain the same. This is a characteristic of the Associative Property, which states that the way in which numbers are grouped when adding does not change their sum.

Step 3: Verify the Associative Property

To verify, observe that in both parts of the equation, we have grouped numbers differently: first as \(2+(x+5)\) and then as \((2+x)+5\). The sum remains the same, which confirms it is an example of the Associative Property of addition.

Key Concepts

Addition

Addition is one of the fundamental operations in mathematics. It involves combining numbers to get a total or sum. In the context of the Associative Property, addition plays a central role. The property shows that when adding numbers, the way the numbers are grouped does not affect the overall sum. For instance, if you have three numbers, you can add them as

• first adding the first two and then the last one, like this: \((a+b)+c\)
• or by adding the last two first, and then the first one: \(a+(b+c)\)

Despite the different grouping in the parentheses, the sum remains unchanged, demonstrating that addition is associative.

Algebra

Algebra is a branch of mathematics that uses symbols, often letters, to represent numbers or quantities in formulas and equations. It allows us to generalize mathematical ideas and solve problems beyond simple arithmetic. In algebra, we use variables such as \(x\) to symbolize unknown values or values that can change. This makes it easier to work with complex calculations and rules, like the Associative Property.
In our example, the equation \(2+(x+5)=(2+x)+5\) utilizes algebra by incorporating a variable \(x\). Through algebra, we understand how changing the grouping of numbers impacts the equation, helping us to see that the positioning of the brackets does not alter the sum.

Mathematical Operation

Operations are processes that take one or more values (operands) and perform a calculation. In mathematics, the fundamental operations include addition, subtraction, multiplication, and division. Each operation has specific properties. In the given example, we are exclusively looking at addition.
A better understanding of operations helps in grasping concepts like the Associative Property. Knowing that addition involves finding the sum of operands, one can see that rearranging these operands when adding will not change the resulting sum. This is unlike other operations, such as subtraction or division, where order matters significantly.

• Mathematical operations form the backbone of solving equations and expressions.
• They are used to manipulate numbers or variables systematically.

Equation

An equation is a mathematical statement that asserts the equality of two expressions. It is a fundamental aspect of algebra and is used to represent real-world and theoretical mathematical problems. Equations often include variables, constants, and operations.
The equation \(2+(x+5)=(2+x)+5\) demonstrates the concept of equality through the Associative Property of addition. Its structure shows that regardless of how the numbers are grouped, the equation remains equal on both sides. Equations like this one are solved by examining and understanding the properties of the operations involved, showcasing how an equation serves as a tool for exploring mathematical relationships.

• Equations help in solving for unknown variables.
• They are essential in expressing relationships between numbers and variables.