Three Times R Less Than 15 Equals 6.

Decoding the Equation: Three Times a Number Less Than 15 Equals 6

This article walks through the mathematical equation "three times a number less than 15 equals 6," breaking down its solution step-by-step and exploring the underlying algebraic concepts. We'll examine how to translate the word problem into an algebraic expression, solve for the unknown variable, and verify the solution. Still, understanding this seemingly simple equation provides a foundational understanding of algebraic problem-solving techniques applicable to more complex scenarios. This exploration is perfect for anyone looking to strengthen their algebra skills, whether you're a student refreshing your knowledge or an adult learner exploring the basics of mathematics.

Understanding the Word Problem

The phrase "three times a number less than 15 equals 6" presents a word problem that requires translation into a mathematical equation. Let's dissect it piece by piece:

• "a number": This represents an unknown value, typically denoted by a variable like x, y, or n. For consistency, we'll use x.

• "three times a number": This translates to 3 * x or 3x.

• "less than 15": This indicates subtraction. We subtract 3x from 15, resulting in 15 - 3x.

• "equals 6": This sets the entire expression equal to 6.

Because of this, the complete algebraic equation becomes: 15 - 3x = 6

Solving the Equation: A Step-by-Step Guide

Solving the equation 15 - 3x = 6 involves isolating the variable x to find its value. We'll achieve this using fundamental algebraic operations.

Step 1: Isolate the term with the variable.

Our goal is to get the term with x (which is -3x) by itself on one side of the equation. To do this, we subtract 15 from both sides of the equation:

15 - 3x - 15 = 6 - 15

This simplifies to:

-3x = -9

Step 2: Solve for the variable.

Now, we need to isolate x. Since x is multiplied by -3, we divide both sides of the equation by -3:

-3x / -3 = -9 / -3

This simplifies to:

x = 3

Which means, the solution to the equation 15 - 3x = 6 is x = 3.

Verification of the Solution

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

15 - 3x = 6

15 - 3(3) = 6

15 - 9 = 6

6 = 6

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

The Underlying Algebraic Principles

This seemingly simple equation demonstrates several key algebraic principles:

• Variable Representation: The use of a variable (x) to represent an unknown quantity is fundamental to algebra. It allows us to express relationships between numbers and solve for unknown values.

• Equation Solving: The process of isolating the variable by performing the same operation on both sides of the equation is a core concept in algebra. This maintains the equality and allows us to solve for the unknown.

• Inverse Operations: We used inverse operations to isolate the variable. Subtraction was used to counteract addition, and division was used to counteract multiplication. This concept of using inverse operations is vital for solving a wide range of algebraic equations.

• Order of Operations (PEMDAS/BODMAS): While not explicitly demonstrated in this simple equation, understanding the order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) is crucial for solving more complex algebraic expressions.

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Expanding the Concept: Similar Word Problems

Let's explore variations of this problem to reinforce the understanding:

• "Five more than twice a number is 11." This translates to 2x + 5 = 11. Solving this gives x = 3.

• "Seven less than four times a number is 9." This translates to 4x - 7 = 9. Solving this gives x = 4.

• "The sum of a number and its double is 15." This translates to x + 2x = 15, which simplifies to 3x = 15, giving x = 5.

These examples highlight the versatility of algebraic techniques in solving various word problems. The key is to carefully translate the words into mathematical symbols and then apply appropriate algebraic methods to solve for the unknown.

Applications in Real-World Scenarios

While seemingly abstract, algebraic problem-solving has countless real-world applications:

• Finance: Calculating interest, determining loan repayments, and budgeting all involve algebraic equations.

• Physics: Solving for variables in equations related to motion, force, and energy.

• Engineering: Designing structures, calculating material requirements, and optimizing systems all rely on algebraic principles.

• Computer Science: Developing algorithms and solving computational problems often involve algebraic concepts.

• Everyday Life: Even simple tasks like calculating discounts, splitting bills, or measuring ingredients for a recipe use basic algebraic principles.

Frequently Asked Questions (FAQ)

Q1: What if the equation was 3x - 15 = 6? How would that change the solution?

A1: The equation 3x - 15 = 6 would be solved differently. You would add 15 to both sides, resulting in 3x = 21, then divide by 3 to get x = 7. The order of operations and the position of the variable significantly impact the solution.

Q2: Can this type of equation have more than one solution?

A2: No, this type of linear equation (an equation with a variable raised to the power of 1) can only have one solution. That's why more complex equations (quadratic, cubic, etc. ) can have multiple solutions.

Q3: What are some common mistakes students make when solving these equations?

A3: Common mistakes include: incorrect application of inverse operations (adding instead of subtracting, multiplying instead of dividing), errors in sign manipulation (forgetting to change signs when moving terms across the equals sign), and incorrect order of operations. Careful attention to detail and systematic steps are crucial.

Q4: How can I improve my algebraic problem-solving skills?

A4: Practice is key! Start with simple equations and gradually work towards more complex problems. Use online resources, textbooks, and practice worksheets to reinforce your understanding. Focus on understanding the underlying concepts rather than just memorizing steps. Seek help when needed – don't hesitate to ask teachers, tutors, or classmates for assistance.

Conclusion

The equation "three times a number less than 15 equals 6" may appear simple at first glance, but it serves as a powerful introduction to the world of algebra. Mastering these fundamental algebraic concepts provides a solid foundation for tackling more challenging mathematical challenges in the future. By understanding the process of translating word problems into equations, applying appropriate algebraic techniques, and verifying solutions, we gain valuable skills applicable to a wide range of mathematical and real-world problems. Consistent practice and a focus on understanding the underlying principles are crucial for success in algebra and beyond.