Three Less Than Six Times A Number

Three Less Than Six Times a Number: A Deep Dive into Algebraic Expressions

This seemingly simple phrase, "three less than six times a number," opens a gateway to a world of algebraic expressions, equations, and problem-solving. Understanding how to translate this phrase into mathematical notation is fundamental to algebra and forms the basis for tackling more complex mathematical concepts. This article will not only explain the translation but also delve into its applications, variations, and how to solve problems involving this expression.

Understanding the Phrase: Deconstructing the Problem

Before diving into the algebraic representation, let's break down the phrase "three less than six times a number" step-by-step:

• "A number": This represents an unknown quantity, which we typically represent with a variable, usually 'x' or 'n'. Let's use 'x' for this example.

• "Six times a number": This translates directly to 6 multiplied by x, written as 6x.

• "Three less than": This signifies subtraction. We're subtracting 3 from the previous result (6x).

Therefore, the complete algebraic expression for "three less than six times a number" is 6x - 3.

From Words to Algebra: The Translation Process

The process of converting word problems into algebraic expressions is a crucial skill in mathematics. Here's a systematic approach you can follow:

1. Identify the unknown: Determine the variable that represents the unknown quantity (in this case, "a number").

2. Break down the phrase: Divide the phrase into smaller, manageable parts, identifying keywords like "times" (multiplication), "plus" (addition), "minus" (subtraction), and "divided by" (division).

3. Translate each part: Convert each part of the phrase into its mathematical equivalent.

4. Combine the parts: Assemble the translated parts to form a complete algebraic expression.

Let's apply this process to a slightly more complex example: "Five more than twice the difference between a number and seven."

1. Unknown: The unknown is "a number," represented by 'x'.

2. Breakdown: "Five more than," "twice," "the difference between a number and seven."

3. Translation:

   • "The difference between a number and seven": x - 7
   • "Twice the difference": 2(x - 7)
   • "Five more than": + 5

4. Combination: The complete expression is 2(x - 7) + 5.

Solving Equations Involving the Expression

Once we have the algebraic expression, we can use it to solve various types of equations. Let's say we're given the equation:

6x - 3 = 21

To solve for 'x', we follow these steps:

1. Add 3 to both sides: 6x = 24

2. Divide both sides by 6: x = 4

Therefore, the number is 4. We can check our answer by substituting x = 4 back into the original equation: 6(4) - 3 = 21, which is true.

Real-World Applications: Where This Expression Appears

This seemingly simple algebraic expression has surprising real-world applications. Consider these examples:

• Pricing: A store might offer a discount. "Six times the number of items purchased, less three dollars" could represent a pricing scheme with a bulk discount.

• Geometry: The perimeter of certain shapes might be described using this expression.

• Physics: Many physics formulas involve linear relationships, which can be modeled using expressions like this.

• Finance: Simple interest calculations could involve a similar structure. "Six times the initial investment, less three dollars in fees".

The possibilities are vast, highlighting the versatility of this core algebraic concept.

Variations and Extensions: Exploring Related Concepts

The core expression, 6x - 3, can be modified and extended to create more complex scenarios. Consider these variations:

• "Three less than six times a number, squared": This becomes (6x - 3)²

• "Three less than six times the sum of a number and two": This translates to 6(x + 2) - 3. Notice the use of parentheses to ensure correct order of operations.

• Inequalities: We can also incorporate inequalities. "Three less than six times a number is greater than 15" would be written as 6x - 3 > 15.

These variations demonstrate the adaptability of the foundational expression and how it can be incorporated into more intricate mathematical problems.

Advanced Applications: Incorporating Other Mathematical Concepts

Let's explore how "three less than six times a number" interacts with other mathematical concepts:

• Functions: We can represent this expression as a function: f(x) = 6x - 3. This allows us to evaluate the expression for different values of x and explore its properties.

• Graphs: Plotting this function on a coordinate plane allows us to visualize the relationship between x and f(x). The graph will be a straight line, highlighting the linear nature of the expression.

• Calculus: While this specific expression isn't particularly complex for calculus, the concepts of derivatives and integrals can be applied to more complex variations of it.

Problem-Solving Strategies: Mastering Word Problems

Successfully tackling word problems requires more than just translating the words into an equation; it involves understanding the context and applying strategic problem-solving techniques. Here's a breakdown of a helpful approach:

1. Read Carefully: Thoroughly read and understand the problem statement. Identify the key information and what the problem is asking you to find.

2. Define Variables: Assign variables to represent the unknown quantities in the problem.

3. Translate to Equations: Carefully translate the word problem into a mathematical equation using the defined variables. This is where understanding the core expression is critical.

4. Solve the Equation: Use appropriate algebraic techniques to solve the equation for the unknown variable(s).

5. Check Your Answer: Always check if the solution makes sense in the context of the problem. Does it answer the question being asked?

Let's illustrate this with an example:

Problem: John is three years younger than six times his son's age. If John is 39 years old, how old is his son?

1. Read Carefully: We need to find the son's age.

2. Define Variables: Let 's' represent the son's age.

3. Translate: John's age is 6s - 3. We know John is 39, so the equation is 6s - 3 = 39.

4. Solve: Adding 3 to both sides gives 6s = 42. Dividing by 6 gives s = 7.

5. Check: Six times 7 (the son's age) minus 3 equals 39 (John's age). The solution is correct.

Conclusion: Expanding Your Algebraic Horizons

The seemingly simple phrase, "three less than six times a number," serves as a powerful introduction to the world of algebra. It demonstrates the critical link between verbal descriptions and mathematical notation. Mastering the translation process, understanding its applications in various contexts, and developing strong problem-solving skills are essential for success in algebra and beyond. This deep dive has not only explained the basics but also highlighted the wider implications and potential complexities that arise when exploring this foundational algebraic expression. By understanding these concepts, you are well-equipped to tackle more advanced algebraic problems and confidently apply your mathematical skills to diverse real-world situations.