Simplifying Algebraic Expressions Finding Equivalents For 3 + 4y - Y(9 - 2y) + 5

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Introduction

In the realm of mathematics, particularly in algebra, simplifying expressions is a fundamental skill. It allows us to rewrite complex expressions into a more manageable and understandable form. This article delves into the process of simplifying the algebraic expression 3 + 4y - y(9 - 2y) + 5. We will meticulously walk through each step, ensuring a clear understanding of how to arrive at the correct equivalent expression. Our main objective is to find the expression that is equivalent to the given one, and we will explore each option to determine the correct answer. Understanding equivalent expressions is crucial, as it forms the basis for solving equations and understanding more complex algebraic concepts. This comprehensive guide aims to provide not only the solution but also a thorough explanation of the underlying principles involved in simplifying algebraic expressions. Let’s embark on this journey of mathematical exploration and simplify this expression step by step.

Problem Statement: Identifying the Equivalent Expression

The core of this article revolves around identifying the expression that is equivalent to the algebraic expression:

3 + 4y - y(9 - 2y) + 5

To find the equivalent expression, we need to simplify the given expression by performing the necessary algebraic operations. These operations include distributing the terms, combining like terms, and arranging the terms in a standard form. The standard form for a quadratic expression, which we anticipate in this case due to the presence of the y² term, is ay² + by + c, where a, b, and c are constants. Our goal is to manipulate the given expression to match this form. This process involves careful application of the distributive property and a keen eye for identifying and combining like terms. Each step is crucial in ensuring the final expression is indeed equivalent to the original. By breaking down the simplification process, we can clearly see how each operation contributes to the final form of the expression. So, let's start by addressing the distributive property and then move on to combining like terms.

Step-by-Step Simplification

1. Distribute the Term

The first step in simplifying the expression is to address the term -y(9 - 2y). We need to distribute the -y across the terms inside the parentheses. This involves multiplying -y by both 9 and -2y. Remember, the distributive property states that a(b + c) = ab + ac. Applying this to our expression, we get:

-y * 9 = -9y

-y * -2y = 2y²

Therefore, the term -y(9 - 2y) simplifies to -9y + 2y². Now, we can rewrite the original expression with this simplification:

3 + 4y - 9y + 2y² + 5

This step is crucial because it removes the parentheses and allows us to combine like terms in the subsequent steps. By carefully applying the distributive property, we ensure that the expression remains equivalent to its original form. Now that we have distributed the term, we can move on to the next step, which involves combining the like terms to further simplify the expression.

2. Combine Like Terms

After distributing the term, our expression looks like this:

3 + 4y - 9y + 2y² + 5

Now, we need to identify and combine the like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have the following like terms:

  • Constant terms: 3 and 5
  • y terms: 4y and -9y

Let's combine these like terms:

  • Combining the constant terms: 3 + 5 = 8
  • Combining the y terms: 4y - 9y = -5y

Now, we can rewrite the expression with the combined like terms:

8 - 5y + 2y²

This step significantly simplifies the expression, making it easier to identify the equivalent form. By combining like terms, we reduce the number of terms in the expression while maintaining its value. The next step involves rearranging the terms into the standard form for a quadratic expression, which will help us match our simplified expression with the given options.

3. Arrange in Standard Form

Our simplified expression is currently:

8 - 5y + 2y²

To make it easier to compare with the given options, we need to arrange the terms in the standard form for a quadratic expression, which is ay² + by + c. In this form, the terms are arranged in descending order of their powers. So, we rearrange the terms as follows:

2y² - 5y + 8

Now, the expression is in the standard quadratic form, making it straightforward to identify the equivalent expression among the given options. This step is crucial for clarity and for matching our result with the provided choices. By arranging the terms in standard form, we eliminate any ambiguity and can confidently select the correct answer. With the expression now in standard form, we can proceed to compare it with the given options and determine the correct equivalent expression.

Comparing with the Options

Now that we have simplified the expression to its equivalent form, 2y² - 5y + 8, we need to compare it with the given options to identify the correct answer. The options provided are:

  • A. 8 - 7y
  • B. 2y² - 5y + 8
  • C. -2y² - 5y + 8
  • D. -6y² + 27y + 8

By carefully comparing our simplified expression with each option, we can see that option B, 2y² - 5y + 8, exactly matches our result. The other options have different coefficients or signs, making them incorrect. Option A is a linear expression, while options C and D have different quadratic and linear terms. Therefore, the equivalent expression is clearly option B.

Conclusion: Identifying the Correct Equivalent Expression

In conclusion, after meticulously simplifying the given expression 3 + 4y - y(9 - 2y) + 5, we arrived at the equivalent expression 2y² - 5y + 8. This simplification involved distributing the term -y across the parentheses, combining like terms, and arranging the expression in the standard quadratic form. By comparing our simplified expression with the given options, we confidently identified option B, 2y² - 5y + 8, as the correct answer. This exercise demonstrates the importance of understanding and applying algebraic principles to simplify expressions. The ability to manipulate expressions and identify equivalent forms is a fundamental skill in mathematics, essential for solving equations and tackling more complex problems. This step-by-step solution provides a clear and concise guide to simplifying algebraic expressions, ensuring a solid understanding of the underlying concepts. Mastering these techniques is crucial for success in algebra and beyond. Therefore, understanding how to simplify expressions like this is not just about getting the right answer; it's about building a strong foundation in mathematical thinking.