Solve: 21 Less A Number Equals 2 Times Sum Of Number & 6

by Alex Johnson 57 views

Let's dive into solving a fascinating mathematical problem! This article aims to break down the equation: "Twenty-one less a number is equal to the product of 2 and the sum of the number and 6." We will explore the step-by-step process of translating this statement into an algebraic equation and then finding the solution. Mathematics can seem daunting at first, but with a clear understanding of the fundamentals, even complex problems can be tackled with ease. So, let's embark on this mathematical journey together!

Understanding the Problem Statement

To begin, it's crucial to understand the problem statement thoroughly. Keywords here include "less," "product," and "sum." These words indicate specific mathematical operations that need to be represented algebraically. "Twenty-one less a number" suggests subtraction, "product" implies multiplication, and "sum" signifies addition. Breaking down the sentence into smaller parts helps in translating it into an equation accurately. The ability to dissect such statements is fundamental in mathematics, and it's a skill that grows with practice. Remember, each word carries significance and translates into a specific operation or element within the equation. By recognizing these key terms, we pave the way for accurate equation formation and ultimately, problem-solving.

Translating Words into an Algebraic Equation

Now, let's translate the words into an algebraic equation. We'll represent the unknown "number" with the variable x. "Twenty-one less a number" can be written as 21 - x. The phrase "is equal to" signifies the equals sign (=). The "product of 2 and the sum of the number and 6" translates to 2(x + 6). Combining these, we get the equation: 21 - x = 2(x + 6). This step is crucial as it bridges the gap between verbal expression and mathematical formulation. Accuracy in translation is paramount because an incorrect equation will invariably lead to a wrong solution. Pay close attention to the order of operations and the precise meaning of each word. Mastering this skill allows for the seamless conversion of real-world problems into solvable mathematical expressions.

Step-by-Step Solution

Here's a step-by-step solution to the equation 21 - x = 2(x + 6):

  1. Distribute the 2 on the right side: 21 - x = 2x + 12.
  2. Add x to both sides: 21 = 3x + 12.
  3. Subtract 12 from both sides: 9 = 3x.
  4. Divide both sides by 3: x = 3.

Each step here follows fundamental algebraic principles, ensuring we maintain the equation's balance and move closer to the solution. Distribution eliminates parentheses, making the equation simpler to manipulate. Adding and subtracting the same terms from both sides isolates the variable term. Finally, division solves for x. It’s essential to understand the rationale behind each step rather than memorizing the process. This understanding fosters adaptability and confidence in solving various algebraic equations. Remember, precision at each stage is vital to achieving the correct answer. Practice is key to mastering these steps and enhancing your problem-solving abilities.

Verifying the Solution

Verifying the solution is an essential step. Substitute x = 3 back into the original equation: 21 - 3 = 2(3 + 6). This simplifies to 18 = 2(9), which further simplifies to 18 = 18. Since both sides are equal, our solution x = 3 is correct. Verification is a critical habit in mathematics. It confirms the accuracy of your solution and identifies any potential errors made during the solving process. By plugging the solution back into the original equation, you ensure that the answer satisfies the equation's conditions. This step provides peace of mind and reinforces your understanding of the problem-solving process. Always make time to verify your solutions, as it enhances both your accuracy and confidence in mathematical problem-solving.

Alternative Methods to Solve the Equation

While we've solved the equation using the standard algebraic method, it's worth noting that there might be alternative approaches. For example, one could graph the two sides of the equation as separate functions and find the point of intersection. Although this method may not be the most efficient for this particular problem, it demonstrates the versatility of mathematical tools. Exploring different problem-solving techniques can deepen understanding and improve critical thinking skills. Alternative methods often offer unique insights and can be particularly useful for more complex problems where standard approaches might be cumbersome. Remember, mathematics is not a rigid set of rules but a flexible toolkit for solving problems. By embracing diverse methods, we not only enhance our problem-solving abilities but also appreciate the richness and interconnectedness of mathematical concepts.

Common Mistakes to Avoid

When solving equations like this, common mistakes often arise from misunderstanding the order of operations or incorrectly distributing numbers. For instance, failing to distribute the 2 correctly in 2(x + 6) could lead to an incorrect equation. Another mistake is making errors in basic arithmetic operations during simplification. Attention to detail is crucial to avoid these pitfalls. Being mindful of the correct order of operations (PEMDAS/BODMAS) is paramount. Double-checking each step can help catch arithmetic errors before they compound. Furthermore, practice with similar problems can help reinforce correct techniques and reduce the likelihood of mistakes. Recognizing these common errors and actively working to avoid them will significantly improve your accuracy and problem-solving efficiency.

Real-World Applications of Algebra

The ability to translate and solve algebraic equations is not just an academic exercise; it has numerous real-world applications. From calculating finances to designing structures, algebra is an indispensable tool. Engineers use it to determine stress and strain in materials, economists use it to model market behavior, and even everyday tasks like budgeting involve algebraic thinking. Understanding the practical applications of algebra makes the subject more engaging and relevant. It highlights the power of mathematics in solving problems we encounter in our daily lives and in various professional fields. By recognizing the connection between algebra and real-world scenarios, we appreciate its importance and are motivated to develop our skills further. Algebra, in essence, is a language that helps us describe and solve the world around us.

Conclusion

In conclusion, solving the equation "Twenty-one less a number is equal to the product of 2 and the sum of the number and 6" involves understanding the problem statement, translating it into an algebraic equation, solving the equation step-by-step, and verifying the solution. Remember, mathematics is a journey of learning and discovery. Keep practicing, and you'll become more confident in your problem-solving abilities. By breaking down complex problems into manageable steps, we can unlock the power of mathematics to solve them. Embrace challenges, seek out different perspectives, and never stop exploring the fascinating world of mathematical equations. Solving this kind of problem is a very helpful exercise to develop our logical and analytical thinking. If you're eager to expand your math skills further, check out resources like Khan Academy for comprehensive lessons and practice exercises.