Problem 7 - Algebra and Trigonometry

Question

\(3-10=\) State the property of real numbers being used. $$ (5 x+1) 3=15 x+3 $$

Step-by-Step Solution

Answer

1. Additive inverse: \(3-10=-7\). 2. Distributive property: \((5x+1)3=15x+3\).

Step 1: Solve the Simple Equation

First, let's solve the given simple equation: \(3-10\). Subtracting gives \(-7\). Thus, \(3 - 10 = -7\).

Step 2: Identify the Property

The operation \(3 - 10 = -7\) involves subtraction, which can relate to the property of additive inverses in real numbers. This property states that for any number \(a\), there exists a number \(-a\) such that \(a + (-a) = 0\). Here, \(-10\) acts as the additive inverse of \(10\).

Step 3: Understand the Second Equation

The equation \((5x + 1)3 = 15x + 3\) involves distributing the multiplication over addition. This showcases the distributive property of multiplication over addition. The distributive property states that for all real numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. In this case, multiplying \(3\) with \(5x + 1\) gives \(15x + 3\).

Key Concepts

Additive Inverses

Additive inverses play a crucial role in simplifying equations and understanding subtraction. When we talk about the additive inverse of a number, we are essentially referring to another number that, when added to our original number, results in zero. This is a foundational aspect of real numbers.

Here's how it works:

• If you have a number like 10, the additive inverse would be -10.
• For any number \(a\), there exists an additive inverse \(-a\) such that \(a + (-a) = 0\).
• This property is what allows subtraction to be interpreted as the addition of an inverse.

In the original exercise, we observed this property when subtracting 10 from 3, resulting in -7. Although it looks like subtraction, it's actually adding the inverse: \(3 + (-10)\), which gives \(-7\). This reveals the elegance of using additive inverses to simplify arithmetic operations.

Distributive Property

The distributive property is a powerful concept in mathematics that combines multiplication and addition. It allows us to multiply a single term by each part of a sum or difference within parentheses. This is particularly useful across algebraic expressions and equations.

The distributive property states that for any real numbers \(a\), \(b\), and \(c\), the following equation holds true:

• \(a(b + c) = ab + ac\)

To illustrate, let's look at the equation from the exercise: \((5x + 1)3 = 15x + 3\). According to the distributive property:

• Multiply \(3\) by \(5x\) to get \(15x\).
• Then, multiply \(3\) by \(1\) to get \(3\).
• When these products are added together, they form \(15x + 3\).

This application of the distributive property helps in expanding expressions and solving equations efficiently.

Real Numbers

Real numbers form the backbone of mathematical calculations. They encompass all the numbers we normally use, including both rational and irrational numbers, providing a comprehensive set for performing arithmetic operations.

Some key points about real numbers include:

• Real numbers include natural numbers, whole numbers, integers, rational numbers (like fractions), and irrational numbers (such as \(\pi\) and \(\sqrt{2}\)).
• They can be positive, negative, or zero.
• They are represented on a continuous number line, forming a dense set where between any two real numbers, there exists another real number.

Understanding real numbers is essential because they provide the context in which properties like additive inverses and the distributive property are defined and applied. Whether we are adding, subtracting, multiplying, or dividing, these properties help us navigate through operations involving real numbers efficiently.