5 Less Than The Product Of 3 And A Number.

5 Less Than the Product of 3 and a Number: A Deep Dive into Algebraic Expressions

This seemingly simple phrase, "5 less than the product of 3 and a number," hides a wealth of mathematical concepts applicable across various fields. This article will explore the meaning of this phrase, translate it into algebraic expressions, solve related equations, and delve into its practical applications. We'll even examine how this foundational concept builds towards more complex algebraic manipulations and problem-solving.

Understanding the Components

Before tackling the complete phrase, let's break down its individual parts:

1. "A Number": The Variable

The phrase "a number" represents an unknown quantity. In algebra, we represent unknown quantities with variables, typically using letters like x, y, or n. For this article, let's use x to represent "a number."

2. "The Product of 3 and a Number": Multiplication

"The product" signifies the result of multiplication. Therefore, "the product of 3 and a number" translates to 3 multiplied by x, which is written algebraically as 3x.

3. "5 Less Than": Subtraction

"5 less than" indicates subtraction. We are subtracting 5 from the previous result. This means we are taking the product of 3 and x (which is 3x) and subtracting 5 from it.

Translating into an Algebraic Expression

Combining the components, we arrive at the algebraic expression for "5 less than the product of 3 and a number":

3x - 5

This concise expression encapsulates the entire phrase. It's crucial to understand the order of operations; multiplication (3x) is performed before subtraction (-5). Changing the order would result in a completely different expression and outcome.

Solving Equations Involving the Expression

The expression 3x - 5 forms the foundation for various equations. Let's examine a few examples:

Example 1: Finding the Number when the Expression Equals 10

Let's say the expression "5 less than the product of 3 and a number" equals 10. This translates to the equation:

3x - 5 = 10

To solve for x, we follow these steps:

1. Add 5 to both sides: 3x - 5 + 5 = 10 + 5 => 3x = 15
2. Divide both sides by 3: 3x / 3 = 15 / 3 => x = 5

Therefore, if "5 less than the product of 3 and a number" equals 10, then the number is 5.

Example 2: Finding the Number when the Expression is Negative

Let's consider a scenario where the expression equals -2:

3x - 5 = -2

Solving for x:

1. Add 5 to both sides: 3x - 5 + 5 = -2 + 5 => 3x = 3
2. Divide both sides by 3: 3x / 3 = 3 / 3 => x = 1

In this case, the number is 1.

Example 3: A More Complex Equation

Let's introduce another variable and create a slightly more complex equation:

3x - 5 = 2y + 1

This equation has two variables, x and y. We cannot solve for specific numerical values of x and y without additional information. However, we can rearrange the equation to express one variable in terms of the other. Let's solve for x:

1. Add 5 to both sides: 3x = 2y + 6
2. Divide both sides by 3: x = (2y + 6) / 3

This shows how the original expression can be manipulated within a more complex algebraic context.

Real-World Applications

While seemingly abstract, the concept of "5 less than the product of 3 and a number" has several real-world applications:

1. Calculating Discounts

Imagine a store offering a discount of $5 on items priced at 3 times a base cost. If the base cost is x, the final price after the discount would be 3x - 5.

2. Modeling Profit

A business might have a profit margin of $5 less than 3 times the cost of goods sold. Again, if the cost of goods sold is x, the profit would be 3x - 5.

3. Temperature Conversion (Simplified Example)

While not a direct application, the concept of subtracting from a multiple can be seen in simplified temperature conversions. (Note: This is a highly simplified example and doesn't represent accurate temperature conversion formulas). Imagine a scenario where a temperature is 3 times a base value minus 5 degrees.

4. Geometry Problems

In geometry, certain problems involving areas or perimeters might involve a similar structure. For example, the area of a rectangle could be expressed as 3 times the width minus 5 square units if the width is represented by x.

Expanding the Concept: Further Algebraic Manipulations

The basic expression 3x - 5 serves as a building block for more complex algebraic expressions and equations. We can:

• Add other terms: For example, 3x - 5 + 2y
• Multiply by a constant: For example, 2(3x - 5) = 6x - 10
• Raise to a power: For example, (3x - 5)² which expands to 9x² - 30x + 25 using the FOIL method.
• Include in inequalities: For example, 3x - 5 > 10, which leads to solving inequalities.

These manipulations demonstrate the versatility of the original expression and its role within a broader algebraic framework.

Conclusion: From Simple Phrase to Complex Concepts

The seemingly simple phrase, "5 less than the product of 3 and a number," offers a gateway into the world of algebra. Through its translation into the algebraic expression 3x - 5, we can solve equations, explore real-world applications, and build a foundation for understanding more complex mathematical concepts. Mastering this basic concept enhances problem-solving skills and opens doors to tackling more advanced mathematical challenges across numerous disciplines. Remember that understanding the underlying principles, such as order of operations and variable representation, is key to successfully manipulating and solving algebraic expressions and equations built upon this foundation.