Solving For Equivalent Expressions An Algebraic Approach

In the realm of algebra, simplifying expressions is a foundational skill. This article will delve into the process of identifying equivalent expressions, focusing on a specific example to illustrate the steps involved. We'll break down the given expression, apply the distributive property, combine like terms, and ultimately arrive at the correct equivalent expression. This comprehensive guide aims to provide a clear understanding of the algebraic manipulation required to solve such problems.

The Problem: Finding Equivalent Algebraic Expressions

Our primary task is to determine which expression is equivalent to the following: $(3y - 4)(2y + 7) + 11y - 9$. This requires us to expand the product of the binomials, combine like terms, and simplify the expression to match one of the given options. The options presented are:

• A. $16y - 6$
• B. $6y^2 + 24y - 37$
• C. $9y - 37$
• D. $6y^2 + 11y + 19$

To solve this, we will systematically apply the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) and then combine like terms to simplify the expression.

Step-by-Step Solution: Expanding and Simplifying

Let's start by expanding the product of the two binomials, $(3y - 4)(2y + 7)$. This involves multiplying each term in the first binomial by each term in the second binomial:

1. First: Multiply the first terms in each binomial: $(3y)(2y) = 6y^2$
2. Outer: Multiply the outer terms: $(3y)(7) = 21y$
3. Inner: Multiply the inner terms: $(-4)(2y) = -8y$
4. Last: Multiply the last terms: $(-4)(7) = -28$

So, expanding $(3y - 4)(2y + 7)$ gives us $6y^2 + 21y - 8y - 28$. Now, we can combine the like terms ($21y$ and $-8y$) to simplify this part of the expression:

$6y^2 + 21y - 8y - 28 = 6y^2 + 13y - 28$

Next, we need to add the remaining terms from the original expression, which are $+11y - 9$. Adding these to our simplified expression gives us:

$6y^2 + 13y - 28 + 11y - 9$

Now, we combine like terms again. We have two $y$ terms ($13y$ and $11y$) and two constant terms ($-28$ and $-9$). Combining these yields:

• $13y + 11y = 24y$
• $-28 - 9 = -37$

Therefore, the fully simplified expression is $6y^2 + 24y - 37$.

Identifying the Correct Answer

Now that we have simplified the original expression to $6y^2 + 24y - 37$, we can compare this to the given options:

• A. $16y - 6$
• B. $6y^2 + 24y - 37$
• C. $9y - 37$
• D. $6y^2 + 11y + 19$

By direct comparison, we can see that our simplified expression, $6y^2 + 24y - 37$, matches option B.

Therefore, the correct answer is B. $6y^2 + 24y - 37$.

Deeper Dive: The Importance of Equivalent Expressions

Understanding equivalent expressions is crucial in algebra for several reasons. Equivalent expressions represent the same value for all values of the variable. This means that while they may look different, they are mathematically identical. This concept is vital for solving equations, simplifying complex algebraic problems, and understanding the fundamental properties of algebraic manipulation.

Applications in Solving Equations

When solving equations, we often need to manipulate expressions to isolate the variable. This involves using algebraic operations to create equivalent expressions that are easier to work with. For example, consider the equation:

$(3y - 4)(2y + 7) + 11y - 9 = 0$

As we determined earlier, the left side of this equation is equivalent to $6y^2 + 24y - 37$. So, we can rewrite the equation as:

$6y^2 + 24y - 37 = 0$

This simplified form makes it easier to apply methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula.

Simplifying Complex Algebraic Problems

In more complex algebraic problems, simplifying expressions is often a necessary first step. Complex expressions can be unwieldy and difficult to work with, but by using techniques like the distributive property and combining like terms, we can reduce them to simpler, equivalent forms. This simplification makes the problems more manageable and less prone to errors.

For instance, consider a problem involving rational expressions, which are fractions with polynomials in the numerator and denominator. To add or subtract rational expressions, we often need to find a common denominator. This involves multiplying the numerator and denominator of each fraction by appropriate factors to obtain equivalent fractions with the same denominator. Simplifying these expressions is crucial for solving the problem correctly.

Understanding Fundamental Properties

Working with equivalent expressions helps in understanding and applying fundamental algebraic properties, such as the commutative, associative, and distributive properties. These properties form the foundation of algebraic manipulation and are essential for proving theorems and solving problems in higher-level mathematics.

• Commutative Property: The commutative property states that the order of operations does not affect the result for addition and multiplication. For example, $a + b = b + a$ and $a imes b = b imes a$.
• Associative Property: The associative property states that the grouping of terms does not affect the result for addition and multiplication. For example, $(a + b) + c = a + (b + c)$ and $(a imes b) imes c = a imes (b imes c)$.
• Distributive Property: The distributive property, as we used in our example, states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, $a(b + c) = ab + ac$.

By manipulating expressions and creating equivalent forms, we gain a deeper understanding of how these properties work and how to apply them in various contexts.

Common Mistakes and How to Avoid Them

When simplifying algebraic expressions, several common mistakes can occur. Recognizing these pitfalls and understanding how to avoid them is crucial for accuracy.

Incorrectly Applying the Distributive Property

A common error is misapplying the distributive property. This often happens when students forget to multiply each term inside the parentheses by the term outside. For example, in the expression $3(x + 2)$, a mistake might be to write $3x + 2$ instead of the correct $3x + 6$. Always ensure that every term inside the parentheses is multiplied by the term outside.

Combining Non-Like Terms

Another frequent mistake is combining terms that are not like terms. Like terms are terms that have the same variable raised to the same power. For example, $3x^2$ and $5x^2$ are like terms, but $3x^2$ and $5x$ are not. When simplifying, only combine like terms. For instance, in the expression $2x + 3y + 4x$, the terms $2x$ and $4x$ can be combined to give $6x$, but $3y$ cannot be combined with them.

Sign Errors

Sign errors are also common, especially when dealing with negative numbers. For example, when expanding $(x - 2)(x + 3)$, the product of $-2$ and $3$ is $-6$, not $6$. Pay close attention to the signs when multiplying and adding terms.

Order of Operations Mistakes

Following the correct order of operations (PEMDAS/BODMAS) is essential. Parentheses (or Brackets), Exponents (or Orders), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Failing to adhere to this order can lead to incorrect simplifications. For example, in the expression $2 + 3 imes 4$, multiplication should be performed before addition, so the correct result is $2 + 12 = 14$, not $5 imes 4 = 20$.

Forgetting to Distribute a Negative Sign

When subtracting an entire expression, it's crucial to distribute the negative sign to every term inside the parentheses. For example, $5 - (2x - 3)$ should be simplified as $5 - 2x + 3$, not $5 - 2x - 3$. Neglecting to distribute the negative sign is a common source of errors.

Practice Problems

To solidify your understanding of equivalent expressions, here are a few practice problems:

1. Which expression is equivalent to $(2x + 3)(x - 4) - 5x + 2$?
2. Simplify the expression $4(y^2 - 2y + 1) - 3(y^2 + y - 2)$.
3. Find the equivalent expression for $(a + b)^2 - 2ab$.

Working through these problems will help reinforce the concepts and techniques discussed in this article.

Conclusion

In conclusion, understanding how to identify and simplify equivalent expressions is a fundamental skill in algebra. By mastering the distributive property, combining like terms, and avoiding common mistakes, you can confidently tackle algebraic problems and gain a deeper appreciation for the structure and beauty of mathematics. The correct answer to our initial problem, $(3y - 4)(2y + 7) + 11y - 9$, is indeed $6y^2 + 24y - 37$, which corresponds to option B. Keep practicing, and you'll find that simplifying expressions becomes second nature.