Simplifying 11(21x/14) - 5x/21 A Step-by-Step Guide

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Introduction

In the realm of mathematics, simplifying algebraic expressions is a foundational skill. It not only streamlines calculations but also enhances understanding of more complex mathematical concepts. This article delves into the step-by-step simplification of the algebraic expression 11(21x/14) - 5x/21, providing a comprehensive guide suitable for students and enthusiasts alike. We will break down each step, explaining the underlying principles and techniques involved. By mastering these simplification methods, you'll be well-equipped to tackle a wide array of algebraic problems.

Understanding the Basics of Algebraic Simplification

Before we dive into the specifics of our expression, it's crucial to understand the fundamental principles of algebraic simplification. Algebraic expressions are combinations of variables (represented by letters like x, y, z), constants (numbers), and mathematical operations (+, -, ×, ÷). Simplifying an expression means rewriting it in a more concise and manageable form without changing its value. This often involves combining like terms, applying the distributive property, and performing arithmetic operations.

Key Principles to Remember

  1. Combining Like Terms: Like terms are terms that have the same variable raised to the same power. For instance, 3x and 5x are like terms because they both contain the variable x raised to the power of 1. We can combine these terms by adding or subtracting their coefficients (the numbers in front of the variables). For example, 3x + 5x = 8x.
  2. The Distributive Property: This property states that a(b + c) = ab + ac. It's used to multiply a term by an expression enclosed in parentheses. For instance, 2(x + 3) = 2x + 6.
  3. Order of Operations (PEMDAS/BODMAS): This acronym dictates the order in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Adhering to this order ensures consistent and accurate simplification.
  4. Fractions: When dealing with fractions in algebraic expressions, it's essential to simplify them by finding common denominators and reducing them to their simplest form. We'll see this in action as we simplify our target expression.

With these principles in mind, let's move on to the simplification process.

Step-by-Step Simplification of 11(21x/14) - 5x/21

Now, let's break down the simplification of the expression 11(21x/14) - 5x/21 into manageable steps. Each step will be explained in detail to ensure clarity and understanding.

Step 1: Simplify the Fraction Inside the Parentheses

The first part of our expression is 11(21x/14). To simplify this, we start by focusing on the fraction inside the parentheses: 21x/14. We can simplify this fraction by finding the greatest common divisor (GCD) of 21 and 14, which is 7. Dividing both the numerator and the denominator by 7 gives us:

(21x ÷ 7) / (14 ÷ 7) = 3x/2

So, our expression now becomes 11(3x/2) - 5x/21.

Step 2: Apply the Distributive Property (Multiplication)

Next, we multiply 11 by the simplified fraction 3x/2. This is a straightforward multiplication of a whole number by a fraction:

11 × (3x/2) = (11 × 3x) / 2 = 33x/2

Our expression is now 33x/2 - 5x/21. This step demonstrates the application of the distributive property, albeit in a simplified form, as we are essentially multiplying 11 by the term inside the parentheses.

Step 3: Find a Common Denominator

To subtract the two fractions, 33x/2 and 5x/21, we need a common denominator. The least common multiple (LCM) of 2 and 21 is 42. Therefore, we will convert both fractions to have a denominator of 42.

To convert 33x/2 to a fraction with a denominator of 42, we multiply both the numerator and the denominator by 21:

(33x × 21) / (2 × 21) = 693x/42

Similarly, to convert 5x/21 to a fraction with a denominator of 42, we multiply both the numerator and the denominator by 2:

(5x × 2) / (21 × 2) = 10x/42

Now, our expression is 693x/42 - 10x/42. Finding a common denominator is a critical step in adding or subtracting fractions, ensuring that we are dealing with comparable units.

Step 4: Subtract the Fractions

With a common denominator, we can now subtract the fractions by subtracting the numerators:

(693x/42) - (10x/42) = (693x - 10x) / 42

This simplifies to:

683x/42

Step 5: Simplify the Result (If Possible)

Finally, we check if the fraction 683x/42 can be simplified further. The GCD of 683 and 42 is 1, which means the fraction is already in its simplest form. Therefore, our final simplified expression is:

683x/42

This concludes the step-by-step simplification of the algebraic expression. By following these steps, we have transformed a seemingly complex expression into its most basic form.

Common Mistakes to Avoid When Simplifying Algebraic Expressions

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

1. Incorrectly Combining Like Terms

One of the most frequent errors is combining terms that are not alike. Remember, like terms must have the same variable raised to the same power. For example, 3x and 3x² are not like terms and cannot be combined. Avoid adding or subtracting coefficients of unlike terms. For instance, 2x + 3y cannot be simplified further because x and y are different variables.

2. Misapplying the Distributive Property

The distributive property is a powerful tool, but it needs to be applied correctly. A common mistake is forgetting to distribute the term to all terms inside the parentheses. For example, in the expression 2(x + 3), you must multiply 2 by both x and 3, resulting in 2x + 6. Failing to multiply by all terms can lead to incorrect simplifications.

3. Ignoring the Order of Operations

As mentioned earlier, the order of operations (PEMDAS/BODMAS) is crucial. Neglecting this order can lead to drastically different and incorrect results. For example, in the expression 3 + 2 × 4, multiplication should be done before addition, so the correct answer is 3 + 8 = 11, not 5 × 4 = 20.

4. Errors with Fractions

Fractions can be a source of confusion. Common mistakes include forgetting to find a common denominator before adding or subtracting fractions, or incorrectly simplifying fractions. Always ensure that you have a common denominator before performing addition or subtraction, and reduce fractions to their simplest form by dividing both the numerator and denominator by their GCD.

5. Sign Errors

Pay close attention to signs, especially when dealing with negative numbers and subtraction. A simple sign error can change the entire outcome of the simplification. For instance, -(x - 2) should be distributed as -x + 2, not -x - 2. Double-check your signs at each step to avoid these errors.

6. Skipping Steps

While it might be tempting to skip steps to save time, this can increase the likelihood of making mistakes. Write out each step clearly, especially when dealing with complex expressions. This allows you to track your work and identify any errors more easily. Skipping steps often leads to overlooking crucial details and misapplying rules.

By being aware of these common mistakes and taking the time to work through each step carefully, you can improve your accuracy and confidence in simplifying algebraic expressions.

Real-World Applications of Algebraic Simplification

Algebraic simplification isn't just an academic exercise; it has numerous practical applications in various fields. Understanding how to simplify expressions can help you solve real-world problems more efficiently.

1. Engineering

In engineering, algebraic simplification is crucial for designing structures, circuits, and systems. Engineers often use algebraic equations to model physical phenomena and must simplify these equations to make calculations and predictions. For example, when calculating the stress on a beam or the current in an electrical circuit, engineers use simplified algebraic expressions to arrive at practical solutions.

2. Physics

Physics relies heavily on mathematical models to describe the behavior of the universe. Simplifying algebraic expressions is essential for solving physics problems related to motion, energy, and forces. Whether calculating projectile trajectories or analyzing the behavior of waves, physicists use simplified equations to make accurate predictions and understand complex phenomena.

3. Computer Science

Computer science utilizes algebraic simplification in algorithm design and optimization. Simplifying Boolean expressions, for instance, is vital in designing efficient logic circuits and optimizing code. In data analysis, simplifying algebraic formulas helps in processing large datasets and extracting meaningful insights. Efficient algorithms rely on simplified expressions to minimize computational complexity and execution time.

4. Economics and Finance

Economic and financial models often involve complex algebraic equations. Simplifying these equations helps economists and financial analysts make predictions about market trends, investments, and economic growth. Whether it's calculating compound interest or analyzing supply and demand curves, algebraic simplification plays a critical role in understanding and predicting economic behavior.

5. Everyday Problem Solving

Beyond professional applications, algebraic simplification can be used in everyday problem-solving. From calculating discounts and sales tax to figuring out proportions in recipes, the ability to simplify expressions can make everyday tasks easier and more efficient. For instance, when planning a budget or comparing prices, simplifying algebraic expressions can help you make informed decisions.

6. Optimization Problems

Many real-world problems involve optimization, where the goal is to find the best possible solution. Algebraic simplification is a key tool in solving these problems, whether it's maximizing profit, minimizing costs, or optimizing resource allocation. Simplified expressions allow for easier analysis and identification of optimal solutions.

These examples illustrate the broad applicability of algebraic simplification. By mastering this skill, you'll be better equipped to tackle problems in various fields and make informed decisions in your daily life. The ability to translate complex situations into simplified mathematical models is a valuable asset in both professional and personal contexts.

Conclusion

In conclusion, simplifying the algebraic expression 11(21x/14) - 5x/21 involves a series of steps, each requiring careful attention to detail. From simplifying fractions and applying the distributive property to finding common denominators and combining like terms, each step is crucial in arriving at the final simplified form: 683x/42. Moreover, we've highlighted the importance of avoiding common mistakes and the wide-ranging real-world applications of algebraic simplification. By mastering these techniques, you'll not only improve your mathematical skills but also enhance your problem-solving abilities in various aspects of life. Algebraic simplification is a foundational skill that opens doors to more advanced mathematical concepts and practical applications, making it an invaluable tool in your academic and professional journey.