Six Less Than Six Times A Number Is 12

Six Less Than Six Times a Number is 12: A Comprehensive Guide to Solving Algebraic Equations

This seemingly simple equation, "Six less than six times a number is 12," hides a wealth of mathematical concepts crucial for understanding algebra. This article will delve deep into solving this equation, exploring various methods, explaining the underlying principles, and extending the knowledge to similar problems. We'll also discuss the practical applications of such equations and how they are used in real-world scenarios.

Understanding the Problem

The statement "Six less than six times a number is 12" can be translated directly into an algebraic equation. Let's break it down step-by-step:

• "a number": This represents an unknown value, which we typically denote with a variable, usually 'x'.
• "six times a number": This translates to 6x (6 multiplied by x).
• "six less than six times a number": This means we subtract 6 from 6x, resulting in 6x - 6.
• "is 12": This indicates that the expression 6x - 6 is equal to 12.

Therefore, the algebraic equation is: 6x - 6 = 12

Solving the Equation: Step-by-Step Guide

We can solve this equation using several methods. Let's explore two common approaches:

Method 1: Using the Addition and Division Properties of Equality

This method relies on the fundamental principles of algebra: whatever operation you perform on one side of the equation, you must perform on the other side to maintain balance.

1. Isolate the term with 'x': Add 6 to both sides of the equation: 6x - 6 + 6 = 12 + 6 This simplifies to: 6x = 18

2. Solve for 'x': Divide both sides by 6: 6x / 6 = 18 / 6 This simplifies to: x = 3

Therefore, the solution to the equation is x = 3. This means the number we were looking for is 3.

Method 2: Using the Distributive Property (Less Common, but Illustrative)

While less efficient for this specific equation, understanding the distributive property is essential for more complex algebraic problems. We can rewrite the equation slightly to demonstrate this method.

Let's consider the equation as: 6(x - 1) = 12

1. Apply the distributive property: The distributive property states that a(b + c) = ab + ac. In our case, we have: 6x - 6 = 12

2. Solve as in Method 1: Following the steps outlined in Method 1, we again arrive at the solution: x = 3

Verifying the Solution

It's crucial to verify our solution by substituting the value of x back into the original equation:

6x - 6 = 12 6(3) - 6 = 12 18 - 6 = 12 12 = 12

Since the equation holds true, our solution x = 3 is correct.

Expanding the Concept: Similar Equations and Problem Solving

The principles applied to solving "Six less than six times a number is 12" are applicable to a wide range of algebraic equations. Let's explore some variations:

Example 1: Five more than three times a number is 20

This translates to the equation: 3x + 5 = 20

Solving this equation:

1. Subtract 5 from both sides: 3x = 15
2. Divide both sides by 3: x = 5

The solution is x = 5.

Example 2: Twice a number minus seven equals nine

This translates to the equation: 2x - 7 = 9

Solving this equation:

1. Add 7 to both sides: 2x = 16
2. Divide both sides by 2: x = 8

The solution is x = 8.

Example 3: Word problems requiring equation formulation

Many real-world problems can be modeled using algebraic equations similar to the one we've discussed. For example:

• Scenario: John earns $6 per hour plus a $12 bonus. If he earned a total of $48, how many hours did he work?

• Equation: 6x + 12 = 48

• Solution:

  1. Subtract 12 from both sides: 6x = 36
  2. Divide both sides by 6: x = 6

John worked 6 hours.

Importance of Understanding Algebraic Equations

The ability to translate word problems into algebraic equations and solve them is a fundamental skill in mathematics and science. It's essential for:

• Problem-solving: Many real-world problems, from calculating finances to designing structures, require solving algebraic equations.
• Data analysis: Understanding equations allows you to analyze data, make predictions, and draw conclusions.
• Further studies: Algebra is the foundation for higher-level mathematics, including calculus, linear algebra, and differential equations.

Beyond the Basics: Exploring More Complex Scenarios

While our initial problem was relatively simple, the principles involved lay the groundwork for solving far more complex algebraic equations. These might involve:

• Multiple variables: Equations with more than one unknown variable.
• Inequalities: Equations involving greater than or less than signs.
• Quadratic equations: Equations involving squared terms (x²).
• Systems of equations: Solving multiple equations simultaneously.

Mastering the fundamental concepts we've explored in this article is crucial for tackling these more advanced topics.

Conclusion: From Simple Equations to Powerful Problem-Solving

The equation "Six less than six times a number is 12," while seemingly straightforward, serves as a powerful introduction to the world of algebra. By understanding the methods for solving this equation and the underlying principles, you've gained a strong foundation for tackling more complex mathematical problems. Remember to practice regularly, explore different problem types, and don't hesitate to seek further resources if needed. With consistent effort, you'll develop the problem-solving skills necessary to confidently approach and solve a wide array of algebraic challenges. The ability to translate real-world situations into mathematical equations is a skill that will serve you well throughout your academic and professional life. So keep practicing, keep learning, and keep exploring the fascinating world of mathematics!