Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ (2 x+3 y)+(x+4 y)=2 x+[3 y+(x+4 y)] $$

Step-by-Step Solution

Answer

The real number property that justifies the statement is the Associative Property of Addition.

Step 1: Identifying the structure of the equation

Observe that the equation is of the form 'A + B = C + D', where: - A: \( (2x + 3y)\) - B: \( (x + 4y)\) - C: \( 2x\) - D: \( [3y + (x + 4y)]\)

Step 2: Check for Commutative property

Since the equation has the same symbols on both sides with the same operation (addition), it is not the Commutative Property.

Step 3: Check for Associative property

The equation is: $$ (2x + 3y) + (x + 4y) = 2x + [3y + (x + 4y)] $$ We can rewrite it as: $$ (A + B) = (C + D) $$ Here, adding D inside the bracket on the right side gives the left side of the equation. So the associativity property is used in the equation. The property is: \(a + ( b + c) = (a + b) + c\)

Step 4: Identify the property

The property used in the given equation is the Associative Property of Addition. So, the real number property that justifies the statement is the Associative Property of Addition.

Key Concepts

Real Number Properties

Understanding the properties of real numbers is fundamental to algebra and applied mathematics. These properties serve as the building blocks for solving equations and simplifying algebraic expressions. Among these, the Associative Property of Addition states that when adding three or more real numbers, the way in which the numbers are grouped does not affect the sum. Mathematically, it is expressed as
\[(a + b) + c = a + (b + c)\]
This property allows us to rearrange and group numbers without changing the result, making calculations more manageable.
In the context of the exercise, applying this property provides clarity on how different components of the expression can be grouped and still result in the same sum. It's important to note that this property holds true for addition of real numbers, which include rational and irrational numbers, integers, and whole numbers. Frequent application of this property is what ultimately builds a strong foundation for more complex mathematical problem solving.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They represent relationships and are used to solve problems in algebra. In our exercise, we have an algebraic expression
\[(2x + 3y) + (x + 4y)\]
which combines variables - in this case, 'x' and 'y' - with coefficients and the operation of addition. Algebraic expressions can be simplified using various properties of real numbers.

The Role of Properties in Simplifying Expressions

By applying the Associative Property of Addition, you can group terms to simplify expressions or solve for variables. This is a key skill for manipulating and understanding algebraic expressions. Simplifying algebraic expressions is like assembling a puzzle; each property you use is a tool that puts pieces together in a clearer, more solvable form. Understanding and identifying the appropriate properties to use with algebraic expressions can vastly improve problem-solving skills in algebra.

Applied Mathematics

Applied mathematics involves applying mathematical theories and principles to solve practical problems from various fields like science, engineering, business, and more. The Associative Property of Addition is often used in applied mathematics to simplify calculations and to reconfigure models by regrouping terms for better computational efficiency.
For example, computational algorithms often use this principle to minimize the number of operations and reduce computational load. Additionally, in financial mathematics, rearranging cash flows or interest calculations utilizes the associative property for structuring complex financial models.

Connecting Theory with Practice

Understanding the real-world implications of mathematical properties such as the Associative Property provides students with a clearer perspective on the importance of these concepts. This property is not just a theoretical rule but a practical tool that can simplify complex problems and enhance analytical strategies in a variety of real-world applications.