Finding The Sum Of Differences In Algebraic Expressions A Step By Step Guide

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In this article, we will delve into the realm of algebraic expressions and explore how to find the sum of their differences. This involves simplifying each expression individually and then combining the results. We'll break down each step with clear explanations and examples, ensuring a solid understanding of the process. This comprehensive guide will cover various algebraic expressions, demonstrating how to handle different terms and coefficients. By the end of this article, you will be well-equipped to tackle similar problems with confidence and precision. Let's dive into the world of algebraic manipulation and master the art of finding the sum of differences.

Introduction to Algebraic Expressions

Before we jump into the problem at hand, let's establish a clear understanding of algebraic expressions. Algebraic expressions are combinations of variables (represented by letters), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division, exponents, etc.). They form the backbone of algebra and are used extensively in various fields of mathematics and science. The ability to manipulate and simplify algebraic expressions is a fundamental skill in mathematics. This involves combining like terms, applying the distributive property, and following the order of operations. Mastering these skills not only helps in solving algebraic equations but also lays a solid foundation for more advanced mathematical concepts. Understanding algebraic expressions is crucial for anyone looking to excel in mathematics or related fields. This section will provide a detailed overview, ensuring you have a strong grasp of the basics before we move on to more complex operations.

Understanding Terms and Coefficients

To effectively work with algebraic expressions, it's essential to understand the concept of terms and coefficients. A term is a single mathematical expression that can be a constant, a variable, or a combination of both, connected by multiplication or division. For example, in the expression 5x^2 + 3x - 2, 5x^2, 3x, and -2 are individual terms. The coefficient is the numerical factor of a term that contains a variable. In the term 5x^2, 5 is the coefficient, and in 3x, 3 is the coefficient. The constant term, such as -2 in our example, is also considered a term but does not have a variable. Recognizing and distinguishing between terms and coefficients is vital for simplifying expressions. It allows us to combine like terms accurately and perform operations such as addition and subtraction correctly. A firm grasp of these concepts is foundational for more advanced algebraic manipulations. This knowledge enables you to break down complex expressions into manageable parts, making the simplification process more straightforward and efficient. Without this understanding, it becomes difficult to correctly manipulate algebraic expressions and solve equations.

Combining Like Terms

One of the fundamental operations in simplifying algebraic expressions is combining like terms. Like terms are terms that have the same variables raised to the same powers. For instance, 3x^2 and 5x^2 are like terms because they both have the variable x raised to the power of 2. However, 3x^2 and 5x are not like terms because the powers of x are different. To combine like terms, we simply add or subtract their coefficients while keeping the variable part the same. For example, 3x^2 + 5x^2 = 8x^2. This process is crucial for simplifying expressions and making them easier to work with. Combining like terms reduces the complexity of the expression, making it clearer and more concise. This skill is essential for solving equations and performing other algebraic manipulations. Mastering the technique of combining like terms allows you to efficiently simplify expressions, which is a cornerstone of algebra. Understanding how to identify and combine like terms will greatly enhance your ability to solve algebraic problems.

Problem Breakdown: Sum of Differences

Now, let's tackle the problem at hand: finding the sum of the differences of the given algebraic expressions. We have five expressions:

  1. −8m2n2+7m2n2−15m2n2{-8m^2n^2 + 7m^2n^2 - 15m^2n^2}
  2. −b2c3+(−b2c3)−(−b2c3){-b^2c^3 + (-b^2c^3) - (-b^2c^3)}
  3. 15m3n−12m2n+8m2n{15m^3n - 12m^2n + 8m^2n}
  4. a2b4+2a2b4−9a2b4{a^2b^4 + 2a^2b^4 - 9a^2b^4}
  5. −18xyz+(−5xyz)−(−12xyz){-18xyz + (-5xyz) - (-12xyz)}

Our task is to simplify each expression individually and then add the simplified expressions together. This involves identifying like terms within each expression, combining their coefficients, and then summing up the results. The process requires careful attention to detail, particularly when dealing with negative signs. By breaking down the problem into smaller steps, we can ensure accuracy and clarity in our solution. Each expression presents its own set of terms and operations, and we will address them one by one. This systematic approach will help us to avoid errors and arrive at the correct final answer. Let's proceed with simplifying each expression step by step.

Step-by-Step Solution

We will now proceed with the step-by-step solution to find the sum of the differences of the given algebraic expressions. This involves simplifying each expression individually and then combining the simplified results. Each step is crucial for ensuring accuracy and clarity in the final answer. We will pay close attention to the signs and coefficients of the terms, applying the rules of algebraic manipulation to arrive at the correct solution. This meticulous approach will demonstrate the process clearly and help you understand how to tackle similar problems in the future. Let’s begin by simplifying the first expression and then proceed sequentially through the remaining expressions.

1. Simplifying the First Expression: −8m2n2+7m2n2−15m2n2{-8m^2n^2 + 7m^2n^2 - 15m^2n^2}

In this expression, all terms are like terms because they all contain the variables m^2 and n^2. To simplify, we add and subtract the coefficients:

−8+7−15=−16{-8 + 7 - 15 = -16}

So, the simplified expression is:

−16m2n2{-16m^2n^2}

This step demonstrates how to combine like terms by adding or subtracting their coefficients. The variables and their exponents remain unchanged as we are simply combining the numerical parts of the terms. This process is a fundamental aspect of simplifying algebraic expressions and is crucial for solving more complex problems. Understanding how to combine like terms efficiently is essential for algebraic manipulation. It allows us to reduce the complexity of expressions, making them easier to work with in subsequent steps. In this case, by combining the coefficients, we have significantly simplified the expression into a single term.

2. Simplifying the Second Expression: −b2c3+(−b2c3)−(−b2c3){-b^2c^3 + (-b^2c^3) - (-b^2c^3)}

Again, we have like terms here. Let's simplify:

−b2c3−b2c3+b2c3{-b^2c^3 - b^2c^3 + b^2c^3}

(−1−1+1)b2c3{(-1 - 1 + 1)b^2c^3}

−b2c3{-b^2c^3}

This expression involves the addition and subtraction of like terms with negative signs. It's crucial to pay close attention to the signs when combining the coefficients. The process involves treating the coefficients as signed numbers and performing the arithmetic operations accordingly. In this case, subtracting a negative term is equivalent to adding a positive term, which is a common point of confusion in algebra. By carefully applying the rules of arithmetic, we can accurately simplify the expression. This step highlights the importance of understanding the rules of sign manipulation in algebraic expressions. Mastering these rules is essential for correctly simplifying and solving algebraic problems.

3. Simplifying the Third Expression: 15m3n−12m2n+8m2n{15m^3n - 12m^2n + 8m^2n}

Here, we have two sets of terms. 15m^3n is not a like term with the others, but -12m^2n and 8m^2n are like terms. Combining the like terms:

−12m2n+8m2n=(−12+8)m2n=−4m2n{-12m^2n + 8m^2n = (-12 + 8)m^2n = -4m^2n}

So, the simplified expression is:

15m3n−4m2n{15m^3n - 4m^2n}

This step demonstrates the process of identifying and combining like terms within a larger expression. Not all terms in an algebraic expression are necessarily like terms, and it's important to distinguish between them. Only like terms can be combined by adding or subtracting their coefficients. This requires careful observation of the variables and their exponents. In this example, we combined the terms -12m^2n and 8m^2n because they both have the same variables raised to the same powers. The term 15m^3n remained separate because it has a different combination of variables and exponents. This skill is crucial for simplifying complex algebraic expressions.

4. Simplifying the Fourth Expression: a2b4+2a2b4−9a2b4{a^2b^4 + 2a^2b^4 - 9a^2b^4}

All terms are like terms in this expression. Combining their coefficients:

(1+2−9)a2b4{(1 + 2 - 9)a^2b^4}

−6a2b4{-6a^2b^4}

This step involves the straightforward combination of like terms. All terms in the expression have the same variables raised to the same powers, making them like terms. The simplification process involves adding and subtracting their coefficients. This example reinforces the importance of paying attention to the signs of the coefficients and performing the arithmetic operations accurately. The result is a single term with a negative coefficient, highlighting the importance of careful calculation. This type of simplification is common in algebra and is a fundamental skill for solving more complex equations and problems.

5. Simplifying the Fifth Expression: −18xyz+(−5xyz)−(−12xyz){-18xyz + (-5xyz) - (-12xyz)}

All terms are like terms. Simplifying:

−18xyz−5xyz+12xyz{-18xyz - 5xyz + 12xyz}

(−18−5+12)xyz{(-18 - 5 + 12)xyz}

−11xyz{-11xyz}

This expression involves the addition and subtraction of like terms with both positive and negative coefficients. It's crucial to correctly apply the rules of sign manipulation to arrive at the correct simplified form. Subtracting a negative term is equivalent to adding a positive term, which is a common point of error in algebraic simplification. By carefully applying these rules, we can combine the coefficients accurately. This step reinforces the importance of understanding and applying the rules of arithmetic in the context of algebraic expressions. The simplified expression is a single term with a negative coefficient, demonstrating the effectiveness of combining like terms.

Summing the Simplified Expressions

Now that we have simplified each expression, we need to find the sum of the results:

  1. −16m2n2{-16m^2n^2}
  2. −b2c3{-b^2c^3}
  3. 15m3n−4m2n{15m^3n - 4m^2n}
  4. −6a2b4{-6a^2b^4}
  5. −11xyz{-11xyz}

Adding these together:

−16m2n2−b2c3+15m3n−4m2n−6a2b4−11xyz{-16m^2n^2 - b^2c^3 + 15m^3n - 4m^2n - 6a^2b^4 - 11xyz}

Combining like terms, we only have like terms with m2n{m^2n}: −16m2n2{-16m^2n^2} and −4m2n{- 4m^2n}

There seems to be a typo because −16m2n2{-16m^2n^2} and −4m2n{- 4m^2n} are not like terms. Thus, the final sum is:

−16m2n2−b2c3+15m3n−4m2n−6a2b4−11xyz{-16m^2n^2 - b^2c^3 + 15m^3n - 4m^2n - 6a^2b^4 - 11xyz}

This final step involves combining the simplified expressions obtained in the previous steps. It requires careful attention to the signs and coefficients of each term. The process involves adding the expressions together and then identifying any like terms that can be further combined. In this case, after adding the simplified expressions, there are no additional like terms to combine. The resulting expression represents the sum of the differences of the original algebraic expressions. This step highlights the importance of accuracy in each preceding step, as any errors made earlier will propagate through to the final answer. The final expression is a combination of several terms with different variables and exponents, representing the complete solution to the problem.

Conclusion

In this article, we have successfully found the sum of the differences of the given algebraic expressions. We accomplished this by systematically simplifying each expression individually and then combining the results. This process involved identifying and combining like terms, paying close attention to the signs and coefficients. By breaking down the problem into smaller, manageable steps, we were able to ensure accuracy and clarity in our solution. This approach is crucial for tackling complex algebraic problems. Mastering the techniques demonstrated in this article will greatly enhance your ability to work with algebraic expressions and solve equations. The key takeaways include understanding the concepts of terms and coefficients, combining like terms, and applying the rules of sign manipulation. These skills are fundamental to success in algebra and will serve as a solid foundation for more advanced mathematical concepts. By practicing these techniques, you can build confidence and proficiency in algebraic problem-solving.