Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ u(3 v+w)=(3 v+w) u $$

Step-by-Step Solution

Answer

The real number property that justifies the statement \(u(3v+w)=(3v+w)u\) is the Commutative Property of Multiplication.

Step 1: Let \(u\), \(v\), and \(w\) represent real numbers. #Step 2: Write down the given statement#

We are given that: \(u(3v+w)=(3v+w)u\) #Step 3: Apply the Commutative Property of Multiplication#

Step 2: The Commutative Property of Multiplication states that for any real numbers \(a\) and \(b\), \(a \times b = b \times a\). We can apply this property to our given statement: #Step 4: Justify the statement#

For the given statement \(u(3v+w)=(3v+w)u\), we can say that the Commutative Property of Multiplication justifies this equality because: 1. \(u\), \(3v\), and \(w\) are real numbers, and 2. The expressions in both sides of the equation are obtained by multiplying the real numbers \(u\) and \((3v+w)\) in different orders. So the property that justifies the statement is the Commutative Property of Multiplication.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics and include all the numbers that can be found on the number line. This means they encompass both rational numbers, like fractions or integers, and irrational numbers, which cannot be expressed as simple fractions. Understanding real numbers is crucial because they form the basis of most mathematical calculations.
Let's look at the types of real numbers:

• **Integers**: Like -3, 0, or 7.
• **Fractions**: Parts of a whole, such as 1/2 or -2/3.
• **Irrational numbers**: Numbers that cannot be written as a simple fraction, like \( \pi \) or \( \sqrt{2} \).
• **Decimals**: Numbers with a decimal point, like 2.5 or 0.333...

In the given exercise, letters \(u\), \(v\), and \(w\) are placeholders for real numbers. This generalization helps us apply properties and theorems, such as the commutative property, across a broad range of numbers.

Multiplication

Multiplication is one of the basic operations in mathematics. It involves combining numbers to find their product. When multiplying numbers, it's crucial to understand certain properties that help us simplify and solve problems.One such property is the **Commutative Property of Multiplication**. This property states:

• For any real numbers \(a\) and \(b\), multiplying them in different orders gives the same result: \(a \times b = b \times a\).

This property is at the heart of the solution to the problem given in the exercise. It allows us to switch the order of multiplication without changing the outcome. So, in our example, \(u(3v + w)\) can be rewritten as \((3v + w)u\), and this transformation uses the commutative property.

Mathematical Justification

Mathematical justification involves proving or explaining why a particular statement, formula, or property holds true. In our exercise, the commutative property of multiplication serves as the justification for rearranging the terms in the expression.Let's break down the process of mathematical justification:

• **Identify the property**: Recognize the relevant property that applies, such as the commutative property of multiplication.
• **Apply the property**: Rearrange or transform the equations or expressions using this property.
• **Explain the reasoning**: Clearly state why the transformation is valid by connecting it to the property or rule.

In our case, the justification comes from realizing that because \(u\), \(v\), and \(w\) are real numbers, the commutative property applies to \(u(3v + w)\) and \((3v + w)u\). This means these two expressions are equivalent, and the property provides a mathematical rationale confirming this equivalence.