Simplifying Algebraic Expressions A Step By Step Guide
This article delves into the process of simplifying the algebraic expression: 2 - (9c + 5d) / 9c + (c + 8d) / 6c. We will meticulously break down each step, ensuring clarity and understanding. This involves finding a common denominator, combining like terms, and ultimately expressing the result in its simplest form. Mastery of simplifying algebraic expressions is crucial in various mathematical fields, including calculus, linear algebra, and differential equations. This guide aims to provide a comprehensive walkthrough, suitable for students and anyone seeking to enhance their algebraic manipulation skills.
Understanding the Components
Before we dive into the simplification process, let's first understand the components of our expression: 2 - (9c + 5d) / 9c + (c + 8d) / 6c. We have a constant term (2), and two fractional terms. The fractional terms involve variables c and d. The denominators are 9c and 6c. The key to simplifying this expression lies in finding a common denominator for the fractional terms, which will allow us to combine them. Understanding the structure and recognizing the different terms is the foundation for a successful simplification. We need to carefully consider the order of operations (PEMDAS/BODMAS) throughout the process. This means we will deal with the fractions before adding or subtracting the constant term. The expression includes both addition and subtraction operations, and the numerators of the fractions are binomials, meaning they consist of two terms. This necessitates distributing any negative signs carefully to avoid errors.
Finding the Least Common Denominator (LCD)
The first step in simplifying our expression, 2 - (9c + 5d) / 9c + (c + 8d) / 6c, is to find the Least Common Denominator (LCD) of the fractions. The denominators are 9c and 6c. To find the LCD, we need to consider the coefficients (9 and 6) and the variable part (c). The Least Common Multiple (LCM) of 9 and 6 is 18. Since both denominators have a factor of c, the LCD will also include c. Therefore, the LCD is 18c. Finding the LCD is a crucial step because it allows us to rewrite the fractions with a common denominator, making it possible to combine them. We achieve this by multiplying each fraction by a form of 1 that results in the desired denominator. Identifying the correct LCD is essential for efficient simplification, as using a larger common denominator would lead to unnecessary complications in later steps. Once we have the LCD, we can proceed to rewrite each fraction with the new denominator.
Rewriting Fractions with the LCD
Now that we've determined the LCD to be 18c, we need to rewrite each fraction in the expression 2 - (9c + 5d) / 9c + (c + 8d) / 6c with this new denominator. For the first fraction, (9c + 5d) / 9c, we need to multiply both the numerator and denominator by 2 to get a denominator of 18c:
[ (9c + 5d) / 9c ] * (2 / 2) = (18c + 10d) / 18c
For the second fraction, (c + 8d) / 6c, we need to multiply both the numerator and denominator by 3 to get a denominator of 18c:
[ (c + 8d) / 6c ] * (3 / 3) = (3c + 24d) / 18c
Finally, we also need to rewrite the constant term 2 with the denominator 18c. This can be done by multiplying 2 by 18c / 18c:
2 * (18c / 18c) = 36c / 18c
Rewriting the terms with a common denominator is a vital step because it allows us to combine the fractions easily. We are essentially expressing each term in a way that allows for direct addition and subtraction of the numerators. This process relies on the fundamental principle that multiplying a fraction by a form of 1 does not change its value.
Combining the Fractions
With all terms now expressed with the common denominator of 18c, we can rewrite the expression 2 - (9c + 5d) / 9c + (c + 8d) / 6c as:
(36c / 18c) - (18c + 10d) / 18c + (3c + 24d) / 18c
Now, we can combine the numerators over the common denominator:
[36c - (18c + 10d) + (3c + 24d)] / 18c
Next, we need to distribute the negative sign in front of the second term:
(36c - 18c - 10d + 3c + 24d) / 18c
Combining the fractions is a significant step, as it consolidates multiple terms into a single fraction. This simplifies the expression and makes it easier to manipulate. The careful distribution of the negative sign is crucial to avoid errors in the subsequent steps. We are now ready to simplify the numerator by combining like terms.
Simplifying the Numerator
Now, let's focus on simplifying the numerator of our expression: (36c - 18c - 10d + 3c + 24d) / 18c. We need to combine the like terms, which are the terms with the variable c and the terms with the variable d.
Combining the c terms:
36c - 18c + 3c = 21c
Combining the d terms:
-10d + 24d = 14d
Therefore, the simplified numerator is 21c + 14d. The expression now becomes:
(21c + 14d) / 18c
Simplifying the numerator is a crucial step towards obtaining the simplest form of the expression. By combining like terms, we reduce the complexity of the expression and make it easier to analyze and manipulate. This step relies on the commutative and associative properties of addition, which allow us to rearrange and group terms. We are now ready to examine the entire fraction for any common factors that can be canceled.
Factoring and Final Simplification
We've reached the expression (21c + 14d) / 18c. Now, we look for common factors in the numerator and denominator to further simplify. The greatest common factor (GCF) of 21 and 14 in the numerator is 7. We can factor out 7 from the numerator:
7(3c + 2d) / 18c
Now, we examine the expression 7(3c + 2d) / 18c for any further common factors between the numerator and the denominator. There are no common factors between 7, (3c + 2d), and 18c. Therefore, this is the simplest form of the expression. Factoring is a powerful technique for simplifying expressions. By identifying and extracting common factors, we can often reduce fractions to their lowest terms. This final step ensures that our answer is presented in the most concise and understandable form. The final simplified expression is a critical result, representing the culmination of the entire simplification process.
Therefore, the simplified form of the expression 2 - (9c + 5d) / 9c + (c + 8d) / 6c is:
7(3c + 2d) / 18c
