Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ (a+2 b)(a-3 b)=a(a-3 b)+2 b(a-3 b) $$

Step-by-Step Solution

Answer

The real number property that justifies the statement \((a+2b)(a-3b)=a(a-3b)+2b(a-3b)\) is the Distributive Property.

Step 1: Identify the Real Number Property

The real number property that justifies the statement is the Distributive Property.

Step 2: Apply the Distributive Property

According to the Distributive Property, for any real numbers a, b, and c, we have \(a(b+c) = ab + ac\). In this exercise, we have: \(a = a\), \(b = 2b\), and \(c = (a-3b)\) Using the Distributive Property, we can rewrite \((a+2b)(a-3b)\) as: \((a+2b)(a-3b) = a(a-3b) + 2b(a-3b)\)

Step 3: Conclusion

The real number property that justifies the given statement is the Distributive Property, which has been successfully applied in this case.

Key Concepts

Real Number Properties

Understanding the properties of real numbers is fundamental to algebra and other areas of mathematics. Real numbers include all the numbers on the number line, encompassing integers, fractions, rational numbers, and irrational numbers. They adhere to specific properties that serve as the foundation for various mathematical operations and proofs.

Some key properties of real numbers include:

• Commutative Property: This property states that you can change the order of the numbers in an addition or multiplication operation without changing the result, such as in \(a+b=b+a\) and \(ab=ba\).
• Associative Property: It tells us that no matter how we group the numbers in an addition or multiplication operation, the result will be the same. For instance, \(a+(b+c)=(a+b)+c\).
• Distributive Property: This property links addition and multiplication. As seen in the exercise, it allows us to multiply a single term by each term inside a parenthesis, exemplified by \(a(b+c)=ab+ac\).
• Identity Property: This pertains to the fact that adding zero or multiplying by one leaves the original number unchanged, represented by \(a+0=a\) and \(a\times1=a\).
• Inverse Property: It holds that every real number has an additive inverse (or opposite) and a multiplicative inverse (or reciprocal) such that when added or multiplied together, they yield the identity elements (0 for addition and 1 for multiplication).

The correct application of these properties is crucial for simplifying algebraic expressions and solving equations.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement. It's a step-by-step demonstration that certain premises lead to a particular conclusion, and it adheres to strict rules of logic to show that under no circumstances can the statement be false.

Proofs can take many forms, like direct proof, indirect proof, proof by contradiction, or constructive proof, each suitable for different kinds of problems. In our context, proving the Distributive Property is involved in demonstrating the validity of operations on real numbers.

The process of proving typically involves:

• Establishing Axioms: These are the accepted truths that do not require proof.
• Defining Theorems: These are statements that can be proved based on axioms and previously proven theorems.
• Logical Deductions: Using a chain of logical reasoning to derive the truth of the statement from the axioms and theorems.

For the given exercise, we can understand that specific steps were taken to validate the statement using the Distributive Property. This step-wise breakdown is a form of proof that strengthens the understanding of the concept at play. Always remember, the finality of a proof lies in its universal acceptance by the mathematical community, leaving no room for doubt about its correctness.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They are the alphabets of the algebraic language and are used to represent real-world situations mathematically. Understanding how to manipulate these expressions using the properties of real numbers is key to solving algebraic problems.

Here are some important aspects to consider about algebraic expressions:

• Terms: An expression is composed of terms, which are the separated parts of an expression that are added or subtracted, like \(a\), \(2b\), or \(a-3b\) in our exercise.
• Coefficients: These are the numerical parts of the terms that multiply the variables, such as the 2 in the term \(2b\).
• Variables: These are symbols representing unknown values that can change, like \(a\) and \(b\) in the expression \(a+2b(a-3b)\).
• Operations: Algebraic expressions include operations like addition, subtraction, multiplication, and sometimes division and exponentiation.

To simplify algebraic expressions, one must be adept at using the properties of real numbers. For instance, applying the Distributive Property allows us to expand expressions and combine like terms, which is exactly what the exercise demonstrates. The realization that algebra is a language and expressions are its sentences will help students navigate mathematical problems with greater comprehension and ease.