Simplifying Algebraic Fractions Expressing 7a/2 - 4/3 As A Single Fraction

In mathematics, simplifying expressions is a fundamental skill. One common task is combining fractions into a single fraction, especially when dealing with algebraic expressions. This article will guide you through the process of expressing the expression ${\frac{7a}{2} - \frac{4}{3}}$ as a single fraction in its simplest form. Understanding how to manipulate fractions is essential for solving more complex algebraic problems, and this step-by-step guide will help you master this skill.

Understanding the Basics of Fraction Subtraction

Before we dive into the specific problem, let's review the basics of subtracting fractions. To subtract two fractions, they must have a common denominator. The common denominator is a multiple of both denominators. Once the fractions have a common denominator, you can subtract the numerators and keep the denominator the same. This principle is crucial for simplifying expressions involving fractions. Think of it like this: you can't directly compare or subtract different "sizes" of fractions until they are expressed in terms of the same "size" (the common denominator). The process involves finding the least common multiple (LCM) of the denominators, which ensures we are using the smallest possible common denominator, thus simplifying our calculations. By adhering to this basic principle, we can confidently tackle more complex fractional expressions.

The common denominator allows us to combine the fractions into a single term, making it easier to work with the expression. In the context of algebraic expressions, this is particularly useful because it allows us to combine like terms and simplify the expression, which is a fundamental skill in algebra. For instance, when solving equations, simplifying expressions often involves combining fractions. Furthermore, in calculus, dealing with complex fractions is a common task, and mastering the basic principles of fraction subtraction is essential for success. Understanding these basic concepts provides a solid foundation for tackling more advanced mathematical problems, ensuring that you can approach complex expressions with confidence and accuracy. The ability to manipulate fractions effectively not only simplifies calculations but also enhances your overall understanding of mathematical concepts.

Finding the Common Denominator for 7a/2 and 4/3

To subtract the fractions ${\frac{7a}{2}}$ and ${\frac{4}{3}}$, the first step is to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators 2 and 3. In this case, the LCM of 2 and 3 is 6. Understanding how to find the LCM is a critical skill in fraction manipulation. The least common multiple is the smallest number that both denominators can divide into evenly. In more complex scenarios, you might need to use prime factorization to find the LCM, but for simple numbers like 2 and 3, it's straightforward.

Once we identify the LCM, we can proceed to rewrite each fraction with the common denominator. This involves multiplying both the numerator and the denominator of each fraction by a factor that will result in the common denominator. This step is crucial because it ensures that the value of the fraction remains the same while changing its form. Without this crucial step, we would be unable to accurately combine the fractions, leading to an incorrect simplification of the original expression. Therefore, mastering the art of finding the LCM and applying it to create equivalent fractions is a cornerstone of algebraic manipulation and simplification.

Rewriting the Fractions with the Common Denominator

Now that we know the common denominator is 6, we need to rewrite both fractions with this denominator. For the fraction ${\frac{7a}{2}}$, we multiply both the numerator and the denominator by 3:

${ \frac{7a}{2} \times \frac{3}{3} = \frac{21a}{6} }$

For the fraction ${\frac{4}{3}}$, we multiply both the numerator and the denominator by 2:

${ \frac{4}{3} \times \frac{2}{2} = \frac{8}{6} }$

By performing these multiplications, we have created equivalent fractions that share a common denominator. This step is critical because it allows us to directly subtract the fractions. Rewriting fractions with a common denominator ensures that we are working with comparable units, much like ensuring you're adding apples to apples rather than apples to oranges. This foundational technique is not only useful for subtraction but also for addition of fractions, making it a cornerstone skill in basic arithmetic and algebra. The process ensures that we maintain the mathematical integrity of the original expression while transforming it into a more manageable form. This meticulous approach prevents errors and lays the groundwork for accurate simplification and problem-solving.

Subtracting the Fractions

Now that both fractions have the common denominator of 6, we can subtract them:

${ \frac{21a}{6} - \frac{8}{6} }$

To subtract fractions with a common denominator, we subtract the numerators and keep the denominator the same:

${ \frac{21a - 8}{6} }$

This is the result of subtracting the two fractions. The numerator, 21a - 8, represents the difference between the two original numerators, and the denominator, 6, remains the same because it is the common unit of measure. Subtracting fractions in this way is a straightforward application of the rules of fraction arithmetic. It's essential to understand that we can only subtract fractions once they share a common denominator, which allows us to accurately determine the difference between their values. The resulting fraction, ${\frac{21a - 8}{6}}$, is a single fraction that represents the original expression. This process not only simplifies the expression but also makes it easier to work with in further calculations or algebraic manipulations. The ability to confidently subtract fractions is a fundamental skill that is crucial for success in higher-level mathematics.

Verifying the Simplest Form

The fraction ${\frac{21a - 8}{6}}$ is now a single fraction. To ensure it is in its simplest form, we need to check if the numerator and denominator have any common factors. In this case, 21a - 8 and 6 do not have any common factors other than 1. Therefore, the fraction is already in its simplest form. Ensuring a fraction is in its simplest form is a crucial step in mathematical simplification. This means that the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be reduced any further. This step often involves identifying the greatest common divisor (GCD) of the numerator and denominator and dividing both by this GCD. However, in the case of ${\frac{21a - 8}{6}}$, the numerator is a linear expression, and the denominator is a constant, making it clear that there are no common factors.

Checking for simplicity is not just a formality; it ensures that the answer is presented in the most concise and clear form. This is particularly important in standardized testing and academic settings where simplified answers are often expected. Moreover, simplified fractions are easier to work with in subsequent calculations, reducing the risk of errors and streamlining the problem-solving process. Therefore, verifying that a fraction is in its simplest form is a fundamental aspect of mathematical practice and demonstrates a thorough understanding of fraction manipulation.

Conclusion

In conclusion, the expression ${\frac{7a}{2} - \frac{4}{3}}$ can be written as a single fraction in its simplest form as ${\frac{21a - 8}{6}}$. This process involves finding the common denominator, rewriting the fractions, subtracting the numerators, and ensuring the resulting fraction is in its simplest form. Mastering these steps is essential for simplifying algebraic expressions and solving mathematical problems involving fractions. Understanding these concepts provides a solid foundation for tackling more advanced mathematical problems. The ability to manipulate fractions effectively not only simplifies calculations but also enhances your overall understanding of mathematical concepts. By following the steps outlined in this article, you can confidently tackle similar problems and improve your algebraic skills.

By practicing these techniques, you'll develop a stronger grasp of fraction manipulation, which is a cornerstone of algebra and higher mathematics. This skill is not only crucial for academic success but also for real-world applications where fractions and ratios are commonly used. Remember, the key to mastering mathematics is consistent practice and a thorough understanding of the fundamental principles. With dedication and effort, you can confidently tackle complex algebraic expressions and achieve your mathematical goals.