Simplifying (67x - 2y) + (3x - 5y) A Comprehensive Guide

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In the realm of mathematics, algebraic expressions serve as the foundation for more complex equations and formulas. Simplifying these expressions is a crucial skill that allows us to manipulate and solve problems effectively. This article delves into the process of simplifying algebraic expressions, focusing on the combination of like terms, using the example expression (67x - 2y) + (3x - 5y) as a practical illustration.

Understanding Algebraic Expressions

Before we dive into the simplification process, it's essential to grasp the fundamental components of algebraic expressions. An algebraic expression comprises variables, coefficients, and constants, connected by mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols, typically letters like 'x' and 'y', that represent unknown quantities. Coefficients are the numerical values that multiply the variables, while constants are standalone numerical values without any variable attached. Like terms are terms that contain the same variable raised to the same power, allowing us to combine them and simplify the expression.

Identifying Like Terms

Identifying like terms is the cornerstone of simplifying algebraic expressions. Like terms possess the same variable raised to the same power. For example, in the expression (67x - 2y) + (3x - 5y), the terms '67x' and '3x' are like terms because they both contain the variable 'x' raised to the power of 1. Similarly, '-2y' and '-5y' are like terms as they both involve the variable 'y' raised to the power of 1. Constants are also considered like terms, as they can be directly combined.

Combining Like Terms: A Step-by-Step Approach

Combining like terms involves adding or subtracting the coefficients of the like terms while keeping the variable and its exponent the same. This process effectively reduces the number of terms in the expression, making it simpler and easier to work with. To illustrate this, let's simplify the expression (67x - 2y) + (3x - 5y) step by step.

  1. Group the like terms: Rearrange the expression to group like terms together. In this case, we can rewrite the expression as (67x + 3x) + (-2y - 5y).
  2. Combine the coefficients: Add or subtract the coefficients of the like terms. For the 'x' terms, we have 67 + 3 = 70. For the 'y' terms, we have -2 - 5 = -7.
  3. Write the simplified expression: Combine the results from the previous step to form the simplified expression. In this case, the simplified expression is 70x - 7y.

Therefore, the simplified form of the expression (67x - 2y) + (3x - 5y) is 70x - 7y. This simplified expression is equivalent to the original expression but contains fewer terms, making it easier to understand and manipulate.

Applying the Principles: Further Examples

To solidify your understanding of simplifying algebraic expressions, let's consider a few more examples:

Example 1: Simplify the expression 5a + 3b - 2a + 4b.

  • Group like terms: (5a - 2a) + (3b + 4b)
  • Combine coefficients: 3a + 7b
  • Simplified expression: 3a + 7b

Example 2: Simplify the expression 2x² - 4x + 3x² + x.

  • Group like terms: (2x² + 3x²) + (-4x + x)
  • Combine coefficients: 5x² - 3x
  • Simplified expression: 5x² - 3x

Example 3: Simplify the expression 7p - 2q + 5 - 3p + q - 8.

  • Group like terms: (7p - 3p) + (-2q + q) + (5 - 8)
  • Combine coefficients: 4p - q - 3
  • Simplified expression: 4p - q - 3

Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is not merely an exercise in mathematical manipulation; it's a fundamental skill with far-reaching implications in various areas of mathematics and beyond. A simplified expression is easier to comprehend, making it less prone to errors during calculations and further manipulations. It also provides a clearer representation of the relationship between variables and constants, facilitating problem-solving and decision-making.

In algebra, simplified expressions are crucial for solving equations, graphing functions, and performing other advanced operations. In calculus, simplifying expressions is often a necessary step before differentiation or integration. In physics and engineering, simplified expressions are used to model physical phenomena and design systems. By mastering the art of simplifying algebraic expressions, you equip yourself with a powerful tool that will serve you well in diverse fields.

Common Mistakes to Avoid

While the process of simplifying algebraic expressions is relatively straightforward, certain common mistakes can hinder accuracy. One frequent error is combining unlike terms. Remember, only terms with the same variable raised to the same power can be combined. For instance, '3x' and '2x²' are unlike terms and cannot be combined.

Another mistake is overlooking the signs of the coefficients. Pay close attention to whether a coefficient is positive or negative, as this will affect the outcome of the addition or subtraction. A common error is mishandling the distributive property when dealing with expressions involving parentheses. Ensure that you correctly distribute the term outside the parentheses to each term inside. For example, in the expression 2(x + 3), you must multiply both 'x' and '3' by 2, resulting in 2x + 6.

To avoid these pitfalls, practice diligently and double-check your work. With careful attention to detail and a solid understanding of the principles, you can confidently simplify algebraic expressions and unlock their full potential.

Conclusion

Simplifying algebraic expressions by combining like terms is a fundamental skill in mathematics with applications extending far beyond the classroom. By mastering this process, you gain the ability to manipulate and solve problems more efficiently. This article has provided a comprehensive guide to simplifying algebraic expressions, covering the identification of like terms, the step-by-step combination process, and the importance of simplification in various mathematical contexts. Remember to practice diligently, avoid common mistakes, and you'll be well on your way to becoming a proficient algebraic simplifier.

Let's dive into the simplification of the algebraic expression (67x - 2y) + (3x - 5y). This seemingly complex expression can be simplified by applying the fundamental principles of combining like terms. In this guide, we will walk you through the process step-by-step, ensuring that you grasp the underlying concepts and can confidently tackle similar problems.

Step 1: Identify Like Terms

The first step in simplifying any algebraic expression is to identify the like terms. As we discussed earlier, like terms are those that contain the same variable raised to the same power. In the expression (67x - 2y) + (3x - 5y), we can identify two pairs of like terms:

  • 67x and 3x: Both of these terms contain the variable 'x' raised to the power of 1.
  • -2y and -5y: Both of these terms contain the variable 'y' raised to the power of 1.

Step 2: Group Like Terms

Once you've identified the like terms, the next step is to group them together. This makes it easier to visualize and combine the terms in the subsequent step. We can rewrite the expression (67x - 2y) + (3x - 5y) as:

(67x + 3x) + (-2y - 5y)

Notice that we've simply rearranged the terms, bringing the like terms together while preserving the original signs (positive or negative) associated with each term.

Step 3: Combine Like Terms

Now comes the crucial step of combining the like terms. To do this, we add or subtract the coefficients of the like terms while keeping the variable and its exponent the same.

  • Combining the 'x' terms: 67x + 3x = (67 + 3)x = 70x
  • Combining the 'y' terms: -2y - 5y = (-2 - 5)y = -7y

We've successfully combined the coefficients of the like terms, resulting in 70x for the 'x' terms and -7y for the 'y' terms.

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the results from the previous step:

70x - 7y

This is the simplified form of the original expression (67x - 2y) + (3x - 5y). It contains only two terms, one involving the variable 'x' and the other involving the variable 'y', making it easier to understand and work with.

Summary of the Simplification Process

To recap, here are the steps involved in simplifying the expression (67x - 2y) + (3x - 5y):

  1. Identify like terms: 67x and 3x, -2y and -5y
  2. Group like terms: (67x + 3x) + (-2y - 5y)
  3. Combine like terms: 70x - 7y
  4. Write the simplified expression: 70x - 7y

By following these steps, you can effectively simplify algebraic expressions involving the addition and subtraction of like terms.

Further Practice

To reinforce your understanding, try simplifying the following expressions using the steps outlined above:

  1. (12a + 5b) - (4a - 2b)
  2. (3x² - 2x + 1) + (x² + 5x - 3)
  3. (8p - 3q + 4) - (2p + q - 6)

The more you practice, the more comfortable and confident you'll become in simplifying algebraic expressions. Remember, the key is to identify like terms, group them together, combine their coefficients, and write the simplified expression.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics that forms the basis for solving more complex equations and problems. This guide has provided a clear and concise explanation of how to simplify the expression (67x - 2y) + (3x - 5y), along with the underlying principles and steps involved. By mastering this skill, you'll be well-equipped to tackle a wide range of algebraic challenges and excel in your mathematical endeavors. Keep practicing, and you'll find that simplifying algebraic expressions becomes second nature.

In the realm of algebra, simplifying expressions is a fundamental skill that unlocks the door to more complex mathematical concepts. This article serves as a comprehensive guide to mastering algebraic simplification, focusing on the specific example of combining the expression (67x - 2y) + (3x - 5y). We will delve into the underlying principles, provide a step-by-step breakdown of the simplification process, and offer valuable insights to enhance your understanding.

The Foundation of Algebraic Simplification

At its core, algebraic simplification aims to rewrite an expression in its most concise and manageable form without altering its mathematical value. This often involves combining like terms, which are terms that share the same variable raised to the same power. For example, in the expression (67x - 2y) + (3x - 5y), '67x' and '3x' are like terms because they both contain the variable 'x' raised to the power of 1. Similarly, '-2y' and '-5y' are like terms as they both contain the variable 'y' raised to the power of 1. Constants, such as numerical values without any variables, are also considered like terms.

Step-by-Step Simplification of (67x - 2y) + (3x - 5y)

Let's embark on a step-by-step journey to simplify the expression (67x - 2y) + (3x - 5y). This methodical approach will not only help you solve this specific problem but also equip you with the skills to tackle similar algebraic challenges.

Step 1: Removing Parentheses

The initial step involves removing the parentheses from the expression. Since we are adding the two expressions within the parentheses, we can simply rewrite the expression without the parentheses, preserving the signs (positive or negative) of each term:

67x - 2y + 3x - 5y

The absence of parentheses makes it easier to identify and group like terms in the subsequent step.

Step 2: Identifying Like Terms

As discussed earlier, like terms are those that share the same variable raised to the same power. In our expression, we can readily identify two pairs of like terms:

  • '67x' and '3x': Both terms contain the variable 'x' raised to the power of 1.
  • '-2y' and '-5y': Both terms contain the variable 'y' raised to the power of 1.

Identifying like terms is the cornerstone of algebraic simplification, as it allows us to combine them and reduce the complexity of the expression.

Step 3: Grouping Like Terms

To facilitate the combination of like terms, we group them together. This can be achieved by rearranging the expression, placing like terms adjacent to each other while preserving their original signs:

67x + 3x - 2y - 5y

Grouping like terms provides a visual aid, making it easier to combine their coefficients in the next step.

Step 4: Combining Like Terms

The heart of algebraic simplification lies in combining like terms. This involves adding or subtracting the coefficients of the like terms while keeping the variable and its exponent unchanged.

  • Combining 'x' terms: 67x + 3x = (67 + 3)x = 70x
  • Combining 'y' terms: -2y - 5y = (-2 - 5)y = -7y

By combining the coefficients of the like terms, we effectively reduce the number of terms in the expression, making it simpler and more manageable.

Step 5: Writing the Simplified Expression

The final step involves writing the simplified expression by combining the results obtained in the previous step. We combine the simplified 'x' term (70x) and the simplified 'y' term (-7y) to arrive at the final simplified expression:

70x - 7y

This is the simplified form of the original expression (67x - 2y) + (3x - 5y). It contains only two terms, making it easier to understand, manipulate, and use in further calculations.

Common Pitfalls to Avoid

While the process of simplifying algebraic expressions is relatively straightforward, certain common pitfalls can lead to errors. Being aware of these potential pitfalls can help you avoid them and ensure accurate simplification.

Combining Unlike Terms

A frequent mistake is combining unlike terms. Remember, only terms with the same variable raised to the same power can be combined. For example, '70x' and '-7y' are unlike terms and cannot be combined.

Ignoring Signs

Pay close attention to the signs (positive or negative) of the coefficients when combining like terms. A negative sign in front of a term indicates that the term is being subtracted, and this must be taken into account during the combination process.

Errors in Arithmetic

Careless arithmetic errors can easily derail the simplification process. Double-check your calculations, especially when dealing with larger numbers or negative coefficients.

The Significance of Algebraic Simplification

Algebraic simplification is not merely a mathematical exercise; it's a fundamental skill with wide-ranging applications. A simplified expression is easier to comprehend, making it less prone to errors during calculations and further manipulations. It also provides a clearer representation of the relationship between variables and constants, facilitating problem-solving and decision-making.

In various fields, such as physics, engineering, and economics, simplified algebraic expressions are used to model real-world phenomena and make predictions. Mastering algebraic simplification equips you with a powerful tool for tackling complex problems and gaining a deeper understanding of the world around you.

Conclusion

Mastering algebraic simplification is an essential step in your mathematical journey. This article has provided a comprehensive guide to simplifying the expression (67x - 2y) + (3x - 5y), along with the underlying principles and steps involved. By understanding the concept of like terms, following the step-by-step simplification process, and avoiding common pitfalls, you can confidently tackle a wide range of algebraic challenges. Remember, practice is key to mastering any mathematical skill, so continue to hone your abilities and unlock the power of algebraic simplification.