How To Simplify The Expression $3y - \frac{(2y - 3)}{4}$ A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of simplifying the algebraic expression $3y - \frac{(2y - 3)}{4}$. Simplifying algebraic expressions is a fundamental skill in mathematics, crucial for solving equations, understanding mathematical concepts, and tackling more complex problems. This article aims to provide a step-by-step breakdown of the simplification process, ensuring clarity and understanding for readers of all levels. We will cover the basic principles of algebraic manipulation, focusing on how to handle fractions, combine like terms, and arrive at the most simplified form of the given expression. By the end of this discussion, you will not only be able to simplify this specific expression but also gain a broader understanding of how to approach similar algebraic problems. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will offer valuable insights and practical techniques. So, let’s embark on this mathematical journey and simplify $3y - \frac{(2y - 3)}{4}$ together.

Understanding the Expression

Before we begin the simplification, it's essential to understand the structure of the expression $3y - \frac{(2y - 3)}{4}$. This expression involves a variable, $y$, and combines both a whole term and a fractional term. The first term, $3y$, is a simple algebraic term where the variable $y$ is multiplied by the constant 3. The second term, $\frac{(2y - 3)}{4}$, is a fraction. The numerator of this fraction is the expression $(2y - 3)$, which means 2 times $y$ minus 3, and the denominator is 4. Understanding fractions and their operations is crucial in simplifying such expressions. In this context, the fraction represents a division of the entire numerator by 4. The minus sign between the two terms indicates that we are subtracting the fractional term from the whole term. To simplify this expression effectively, we need to address the fraction first and then combine like terms. This involves applying the principles of order of operations and algebraic manipulation, which we will discuss in detail in the following sections. Recognizing the components of the expression is the first step towards simplifying it, allowing us to develop a strategy for tackling the problem systematically.

Step-by-Step Simplification Process

1. Eliminating the Fraction

The first critical step in simplifying the expression $3y - \frac{(2y - 3)}{4}$ is to eliminate the fraction. To do this, we need to find a common denominator for all terms in the expression. In this case, the whole term $3y$ can be considered as $\frac{3y}{1}$, and the fractional term is $\frac{(2y - 3)}{4}$. The least common denominator (LCD) for 1 and 4 is 4. Therefore, we multiply both the numerator and the denominator of the term $3y$ by 4 to get a common denominator. This transforms the expression as follows:

$$\frac{3y}{1} \times \frac{4}{4} - \frac{(2y - 3)}{4}$$

This gives us:

$$\frac{12y}{4} - \frac{(2y - 3)}{4}$$

Now that both terms have the same denominator, we can combine them. This process of eliminating the fraction is crucial because it allows us to work with a single, unified expression, making subsequent steps more straightforward. By converting the whole term into a fraction with the common denominator, we set the stage for combining the numerators and simplifying the expression further.

2. Combining the Numerators

Now that we have a common denominator, we can combine the numerators of the two fractions. The expression is now in the form:

$$\frac{12y}{4} - \frac{(2y - 3)}{4}$$

To combine these fractions, we subtract the second numerator from the first. It’s crucial to handle the subtraction carefully, especially since the second numerator is an expression in parentheses. We distribute the negative sign across the terms inside the parentheses:

$$\frac{12y - (2y - 3)}{4}$$

Distributing the negative sign, we get:

$$\frac{12y - 2y + 3}{4}$$

Notice how the $-3$ inside the parentheses becomes $+3$ when we subtract the entire expression. This step of combining the numerators is a key algebraic manipulation. It brings us closer to simplifying the expression by consolidating the terms into a single fraction. The correct application of the distributive property here is essential to avoid errors and maintain the integrity of the mathematical statement. Once the numerators are combined, we can focus on simplifying the resulting expression in the numerator.

3. Simplifying the Numerator

After combining the numerators, our expression looks like this:

$$\frac{12y - 2y + 3}{4}$$

The next step is to simplify the numerator by combining like terms. In this case, the like terms are $12y$ and $-2y$. Combining like terms is a fundamental algebraic operation that involves adding or subtracting terms that have the same variable raised to the same power. Here, both terms have the variable $y$ raised to the power of 1, so we can combine them:

$$12y - 2y = 10y$$

So, the numerator simplifies to:

$$10y + 3$$

Now, our expression looks like this:

$$\frac{10y + 3}{4}$$

The numerator is now in its simplest form, as there are no more like terms to combine. This step is crucial because it reduces the complexity of the expression, making it easier to understand and work with. By simplifying the numerator, we are one step closer to the final simplified form of the entire expression. In the next step, we will examine if further simplification is possible.

4. Final Simplified Form

After simplifying the numerator, we have the expression:

$$\frac{10y + 3}{4}$$

Now, we need to determine if this is the final simplified form. To do this, we check if there are any common factors between the numerator and the denominator that can be canceled out. In this case, the numerator is $10y + 3$, and the denominator is 4. The terms in the numerator, $10y$ and $3$, do not have any common factors other than 1, and neither does the denominator 4. This means we cannot simplify the fraction any further by canceling out common factors. Therefore, the expression $\frac{10y + 3}{4}$ is the final simplified form. We can also express this as the sum of two fractions:

$$\frac{10y}{4} + \frac{3}{4}$$

Which simplifies to:

$$\frac{5y}{2} + \frac{3}{4}$$

Both $\frac{10y + 3}{4}$ and $\frac{5y}{2} + \frac{3}{4}$ are acceptable simplified forms of the original expression. This final step is crucial in ensuring that we have reached the most reduced and understandable form of the expression. By verifying that no further simplification is possible, we complete the process and present the answer in its most concise form. The final simplified form represents the original expression in its most basic terms, making it easier to use in further mathematical operations or analyses.

Alternative Approaches

While the step-by-step method described above is a straightforward way to simplify the expression $3y - \frac{(2y - 3)}{4}$, there are alternative approaches that one might consider. One such approach involves distributing the negative sign in front of the fraction before finding a common denominator. This can be particularly useful for those who find it easier to work with terms individually rather than within a fraction. Here’s how this alternative approach would work:

1. Distribute the Negative Sign: First, distribute the negative sign across the terms in the numerator of the fraction:

   $$3y - \frac{(2y - 3)}{4} = 3y - \frac{2y}{4} + \frac{3}{4}$$

   Notice that the $-3$ inside the parentheses becomes $+3$ when the negative sign is distributed.

2. Simplify the Fraction: Next, simplify the fraction $\frac{2y}{4}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

   $$\frac{2y}{4} = \frac{y}{2}$$

   Now the expression looks like:

   $$3y - \frac{y}{2} + \frac{3}{4}$$

3. Find a Common Denominator: To combine the terms, we need a common denominator. The least common denominator for 1 (from $3y$), 2, and 4 is 4. Convert each term to have this denominator:

   $$\frac{3y \times 4}{4} - \frac{y \times 2}{2 \times 2} + \frac{3}{4} = \frac{12y}{4} - \frac{2y}{4} + \frac{3}{4}$$

4. Combine Like Terms: Now, combine the terms in the numerator:

   $$\frac{12y - 2y + 3}{4} = \frac{10y + 3}{4}$$

This alternative approach yields the same simplified form, $\frac{10y + 3}{4}$. The choice of method often depends on personal preference and what feels most intuitive to the individual. This highlights the flexibility within algebra and how different paths can lead to the same correct answer. Exploring alternative approaches not only reinforces understanding but also enhances problem-solving skills, allowing for a more adaptable and confident approach to mathematical challenges.

Common Mistakes to Avoid

When simplifying algebraic expressions like $3y - \frac{(2y - 3)}{4}$, it’s easy to make mistakes if you’re not careful. Avoiding common mistakes is crucial for accurate simplification. One frequent error is failing to distribute the negative sign correctly when dealing with the subtraction of a fraction. Remember, the negative sign applies to the entire numerator, so you need to distribute it to each term inside the parentheses. For instance, in our expression, it's important to remember that:

$$- (2y - 3) = -2y + 3$$

Another common mistake is incorrectly combining terms. You can only combine like terms—terms with the same variable raised to the same power. For example, $12y$ and $-2y$ can be combined, but $10y$ and $3$ cannot, because 3 is a constant term without a variable. Errors can also occur when finding a common denominator. Make sure you correctly identify the least common denominator and adjust the numerators accordingly. For example, when converting $3y$ to a fraction with a denominator of 4, it should be $\frac{12y}{4}$, not something else. Additionally, be cautious when simplifying fractions. Ensure you’ve simplified as much as possible by checking for common factors between the numerator and the denominator. By being aware of these common pitfalls and double-checking your work, you can increase your accuracy and confidence in simplifying algebraic expressions.

Conclusion

In conclusion, simplifying the expression $3y - \frac{(2y - 3)}{4}$ involves a series of steps that require careful application of algebraic principles. We began by understanding the structure of the expression, identifying the whole term and the fractional term. The first crucial step was eliminating the fraction by finding a common denominator, which allowed us to combine the terms. We then focused on simplifying the numerator by combining like terms, which resulted in a more concise expression. Finally, we verified that the expression was in its simplest form by checking for common factors between the numerator and the denominator. Through this step-by-step process, we arrived at the simplified form, $\frac{10y + 3}{4}$, or alternatively, $\frac{5y}{2} + \frac{3}{4}$. We also explored an alternative approach, demonstrating that different methods can lead to the same correct answer, highlighting the flexibility within algebraic problem-solving. Furthermore, we discussed common mistakes to avoid, emphasizing the importance of distributing negative signs correctly and combining like terms accurately. Mastering the simplification of algebraic expressions is a fundamental skill in mathematics, and by understanding the process and practicing diligently, you can build confidence and proficiency in algebra. The ability to simplify expressions is not only essential for solving equations but also for understanding more advanced mathematical concepts. We hope this comprehensive guide has provided you with the knowledge and tools necessary to tackle similar algebraic challenges with ease and accuracy.