Simplifying The Expression (a+b)/(ab) - (b+c)/(bc) A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the simplification of the algebraic expression (a+b)/(ab) - (b+c)/(bc). This exploration will not only showcase the step-by-step process of simplification but also highlight the underlying mathematical principles and techniques involved. Understanding these techniques is crucial for anyone studying algebra and beyond, as they form the building blocks for more complex problem-solving.

Deconstructing the Expression: (a+b)/(ab) - (b+c)/(bc)

To effectively simplify the expression (a+b)/(ab) - (b+c)/(bc), we first need to break it down into its components and analyze each part. This involves recognizing the structure of the expression, identifying common denominators, and understanding how to manipulate fractions. Let's delve into the intricacies of this process.

The given expression consists of two fractions: (a+b)/(ab) and (b+c)/(bc), subtracted from each other. Each fraction has a binomial numerator and a monomial denominator. The first fraction, (a+b)/(ab), represents the sum of a and b divided by the product of a and b. Similarly, the second fraction, (b+c)/(bc), represents the sum of b and c divided by the product of b and c. To subtract these two fractions, we need to find a common denominator. The common denominator for ab and bc is the least common multiple (LCM) of the two denominators. In this case, the LCM of ab and bc is abc. This is because abc is the smallest expression that is divisible by both ab and bc. Once we have the common denominator, we can rewrite each fraction with this denominator. To rewrite (a+b)/(ab) with the denominator abc, we multiply both the numerator and the denominator by c. This gives us c(a+b)/(abc). Similarly, to rewrite (b+c)/(bc) with the denominator abc, we multiply both the numerator and the denominator by a. This gives us a(b+c)/(abc). Now that both fractions have the same denominator, we can subtract them. This involves subtracting the numerators while keeping the denominator the same. So, we have [c(a+b) - a(b+c)]/(abc). The next step is to expand the numerators by distributing c in the first term and a in the second term. This gives us (ac + bc - ab - ac)/(abc). Notice that the term ac appears with both positive and negative signs in the numerator. These terms cancel each other out, simplifying the expression further. This leaves us with (bc - ab)/(abc). We can factor out a b from the numerator, giving us b(c-a)/(abc). Finally, we can cancel the common factor b from the numerator and the denominator, resulting in the simplified expression (c-a)/(ac). This is the simplest form of the original expression, and it highlights the importance of finding common denominators, expanding terms, and canceling common factors in algebraic simplification.

Step-by-Step Simplification of (a+b)/(ab) - (b+c)/(bc)

To effectively simplify mathematical expressions, a methodical approach is essential. The simplification of (a+b)/(ab) - (b+c)/(bc) follows a precise sequence of steps, each grounded in fundamental algebraic principles. Let's meticulously dissect the step-by-step process involved.

  1. Finding a Common Denominator: The initial hurdle in subtracting fractions is to ensure they share a common denominator. In the given expression, the denominators are ab and bc. To find the least common denominator (LCD), we identify the least common multiple (LCM) of ab and bc. The LCM of ab and bc is abc. This is because abc is the smallest expression that is divisible by both ab and bc. The importance of finding the LCD lies in the fact that it allows us to combine the fractions into a single fraction, making the subsequent steps of simplification much easier. Without a common denominator, we cannot directly subtract the fractions. The concept of the least common multiple is not only crucial in algebraic simplification but also in various other areas of mathematics, such as number theory and arithmetic operations involving fractions.

  2. Rewriting Fractions with the Common Denominator: Once the common denominator abc is identified, each fraction needs to be rewritten with this new denominator. To rewrite the first fraction, (a+b)/(ab), with the denominator abc, we multiply both the numerator and the denominator by c. This gives us c(a+b)/(abc). Similarly, to rewrite the second fraction, (b+c)/(bc), with the denominator abc, we multiply both the numerator and the denominator by a. This results in a(b+c)/(abc). This step is crucial because it ensures that we are working with equivalent fractions. Multiplying both the numerator and the denominator by the same value does not change the value of the fraction, but it allows us to perform the subtraction. This technique is a fundamental principle in fraction manipulation and is widely used in various mathematical contexts.

  3. Subtracting the Fractions: With both fractions now sharing the common denominator abc, we can proceed with the subtraction. This involves subtracting the numerators while keeping the denominator constant. So, the expression becomes [c(a+b) - a(b+c)]/(abc). This step is a direct application of the rules of fraction subtraction. When subtracting fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. This principle is a cornerstone of arithmetic operations involving fractions and is essential for simplifying complex expressions. The ability to subtract fractions accurately is a fundamental skill in mathematics and is crucial for solving various types of problems.

  4. Expanding the Numerators: The next step involves expanding the numerators by applying the distributive property. In the expression [c(a+b) - a(b+c)]/(abc), we distribute c in the first term and a in the second term. This gives us (ac + bc - ab - ac)/(abc). The distributive property is a fundamental algebraic principle that states that a(b+c) = ab + ac. Applying this property correctly is crucial for expanding expressions and simplifying them. This step allows us to remove the parentheses and combine like terms in the next step. The ability to apply the distributive property is a fundamental skill in algebra and is essential for manipulating algebraic expressions.

  5. Simplifying the Numerator: After expanding the numerators, we simplify the numerator by combining like terms. In the expression (ac + bc - ab - ac)/(abc), we observe that the term ac appears with both positive and negative signs. These terms cancel each other out, simplifying the expression further. This leaves us with (bc - ab)/(abc). Combining like terms is a crucial step in simplifying algebraic expressions. It involves identifying terms that have the same variables raised to the same powers and adding or subtracting their coefficients. This process helps to reduce the complexity of the expression and make it easier to work with. The ability to combine like terms is a fundamental skill in algebra and is essential for solving equations and simplifying expressions.

  6. Factoring the Numerator: To further simplify the expression, we factor the numerator. In the expression (bc - ab)/(abc), we can factor out a common factor of b from the numerator. This gives us b(c-a)/(abc). Factoring is the process of finding the factors of an expression, which are the expressions that multiply together to give the original expression. Factoring is a crucial technique in algebra and is used to simplify expressions, solve equations, and analyze functions. In this case, factoring out the common factor b allows us to simplify the expression further by canceling it with a factor in the denominator.

  7. Canceling Common Factors: The final step in the simplification process is to cancel any common factors between the numerator and the denominator. In the expression b(c-a)/(abc), we can cancel the common factor b from the numerator and the denominator. This results in the simplified expression (c-a)/(ac). Canceling common factors is a fundamental technique in fraction simplification. It involves dividing both the numerator and the denominator by the same factor, which reduces the fraction to its simplest form. This step is crucial for obtaining the most simplified version of the expression. The ability to cancel common factors is a fundamental skill in arithmetic and algebra and is essential for working with fractions and rational expressions.

By meticulously following these steps, we arrive at the simplified form of the expression, (c-a)/(ac). This process underscores the importance of a systematic approach in mathematics, where each step builds upon the previous one to achieve the desired result.

Key Mathematical Principles in Action

The simplification of (a+b)/(ab) - (b+c)/(bc) vividly illustrates the application of several key mathematical principles. These principles are not only fundamental to algebra but also serve as the bedrock for more advanced mathematical concepts. Let's delve into the core principles at play in this simplification process.

  • The Concept of a Common Denominator: The cornerstone of subtracting fractions lies in the concept of a common denominator. Fractions can only be directly added or subtracted if they share the same denominator. This is because the denominator represents the number of equal parts into which a whole is divided, and to combine fractions, we need to ensure that these parts are of the same size. In our expression, the denominators ab and bc are different, necessitating the identification of a common denominator before subtraction. The least common multiple (LCM) serves as the ideal common denominator, as it is the smallest expression divisible by both denominators. In this case, the LCM of ab and bc is abc, which becomes our common denominator. Understanding and applying the concept of a common denominator is crucial not only in algebra but also in arithmetic and various other branches of mathematics. It allows us to perform operations on fractions and combine them into a single, simplified fraction.

  • The Distributive Property: The distributive property is a fundamental algebraic principle that plays a pivotal role in expanding expressions. It states that multiplying a sum by a number is the same as multiplying each addend separately by the number and then adding the products. In our simplification, the distributive property is applied when expanding the numerators of the fractions. For instance, in the expression c(a+b), we distribute c to both a and b, resulting in ac + bc. Similarly, in a(b+c), we distribute a to both b and c, resulting in ab + ac. The distributive property is essential for removing parentheses and simplifying expressions. It allows us to transform expressions into a form where like terms can be combined, leading to further simplification. This property is a cornerstone of algebraic manipulation and is widely used in solving equations, simplifying expressions, and various other mathematical contexts.

  • Combining Like Terms: Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same powers. In our simplification, after expanding the numerators, we encounter the expression (ac + bc - ab - ac)/(abc). Here, we observe that the term ac appears twice, once with a positive sign and once with a negative sign. These like terms cancel each other out, simplifying the expression to (bc - ab)/(abc). Combining like terms involves adding or subtracting the coefficients of these terms while keeping the variable part the same. This process helps to reduce the complexity of the expression and make it easier to work with. The ability to identify and combine like terms is a fundamental skill in algebra and is essential for solving equations, simplifying expressions, and performing various algebraic manipulations.

  • Factoring: Factoring is the process of breaking down an expression into its constituent factors, which are the expressions that multiply together to give the original expression. In our simplification, we factor the numerator (bc - ab) by identifying the common factor b. Factoring out b results in the expression b(c-a). Factoring is a powerful technique in algebra that serves multiple purposes. It can be used to simplify expressions, solve equations, and analyze functions. By factoring an expression, we can often reveal hidden structures and simplify complex expressions into more manageable forms. In our case, factoring the numerator allows us to identify a common factor with the denominator, which can then be canceled, leading to further simplification.

  • Canceling Common Factors: Canceling common factors is a fundamental technique in fraction simplification. It involves dividing both the numerator and the denominator by the same factor, which reduces the fraction to its simplest form. In our simplification, after factoring the numerator, we have the expression b(c-a)/(abc). We observe that both the numerator and the denominator share a common factor of b. Canceling this common factor results in the simplified expression (c-a)/(ac). Canceling common factors is crucial for obtaining the most simplified version of a fraction or a rational expression. It ensures that the numerator and the denominator have no factors in common, making the expression as concise as possible. The ability to identify and cancel common factors is a fundamental skill in arithmetic and algebra and is essential for working with fractions and rational expressions.

By understanding and applying these key mathematical principles, we can effectively simplify the expression (a+b)/(ab) - (b+c)/(bc) and gain a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

Practical Applications and Further Exploration

The simplification of (a+b)/(ab) - (b+c)/(bc) is not merely an academic exercise; it has practical applications in various fields and serves as a gateway to more advanced mathematical concepts. Let's explore the real-world relevance of this simplification and delve into avenues for further learning.

In various scientific and engineering disciplines, algebraic expressions often arise in the formulation of models and the solution of problems. Simplifying these expressions is crucial for making calculations manageable and gaining insights into the underlying phenomena. For instance, in physics, equations describing motion, energy, and forces often involve complex algebraic expressions. Simplifying these expressions can lead to a better understanding of the relationships between different physical quantities. Similarly, in engineering, simplifying expressions is essential for designing structures, analyzing circuits, and optimizing processes. The ability to manipulate and simplify algebraic expressions is a valuable skill for anyone working in these fields.

Furthermore, the techniques used in simplifying (a+b)/(ab) - (b+c)/(bc), such as finding common denominators, applying the distributive property, combining like terms, factoring, and canceling common factors, are not limited to this specific expression. These techniques are widely applicable to a broad range of algebraic expressions and equations. Mastering these techniques provides a solid foundation for tackling more complex mathematical problems. For instance, these techniques are essential for solving quadratic equations, simplifying rational expressions, and working with trigonometric identities. The more comfortable you are with these fundamental algebraic manipulations, the better equipped you will be to handle advanced mathematical concepts.

For those seeking to deepen their understanding and explore further, there are several avenues to pursue. Delving into more complex algebraic expressions and equations can provide valuable practice and challenge your skills. You can explore simplifying expressions involving multiple variables, exponents, and radicals. You can also investigate techniques for solving different types of equations, such as linear equations, quadratic equations, and systems of equations. Additionally, studying related mathematical concepts, such as rational functions, partial fractions, and calculus, can provide a broader perspective and enhance your problem-solving abilities.

Online resources, textbooks, and educational videos offer a wealth of information and guidance for further exploration. Numerous websites provide tutorials, examples, and practice problems on algebraic simplification and related topics. Textbooks on algebra and precalculus offer comprehensive coverage of these concepts, along with detailed explanations and exercises. Educational videos can provide visual demonstrations and step-by-step solutions to complex problems. By utilizing these resources and engaging in continuous learning, you can strengthen your mathematical skills and expand your knowledge.

In conclusion, the simplification of (a+b)/(ab) - (b+c)/(bc) is a valuable exercise that demonstrates the power of fundamental algebraic principles. By mastering these principles and exploring further, you can unlock a world of mathematical possibilities and enhance your problem-solving capabilities.