Simplifying Algebraic Expressions An Equivalent Expression To 6x^3 + 3y^2 - 5x^3 + 2y^2

In this article, we will break down the problem step-by-step and identify the equivalent expression. Let's dive into the solution!

Understanding the Problem

The question presents an algebraic expression: $6x^3 + 3y^2 - 5x^3 + 2y^2$. Our task is to simplify this expression by combining like terms and determining which of the provided options (A, B, C, or D) is equivalent to the simplified form. This involves understanding the basic principles of algebraic manipulation, particularly the combination of terms with the same variables and exponents. This is a fundamental concept in algebra, often encountered in introductory courses and standardized tests.

The ability to simplify algebraic expressions is crucial in various mathematical contexts, including solving equations, graphing functions, and understanding more advanced algebraic concepts. Simplifying expressions makes them easier to work with and interpret. In this case, we need to identify and combine like terms, which are terms that have the same variable raised to the same power. For example, $6x^3$ and $-5x^3$ are like terms because they both contain the variable $x$ raised to the power of 3. Similarly, $3y^2$ and $2y^2$ are like terms because they both contain the variable $y$ raised to the power of 2.

To solve this problem effectively, we need to apply the commutative and associative properties of addition. The commutative property allows us to rearrange the terms in the expression without changing its value (e.g., $a + b = b + a$). The associative property allows us to group terms in different ways without changing the value (e.g., $(a + b) + c = a + (b + c)$). By rearranging and grouping like terms, we can simplify the expression and easily identify the correct answer. Errors in simplifying algebraic expressions often arise from incorrectly combining unlike terms or misapplying the rules of arithmetic with signed numbers. Therefore, careful attention to detail and a solid understanding of algebraic principles are essential for solving this type of problem accurately.

Step-by-Step Solution

To find the expression equivalent to $6x^3 + 3y^2 - 5x^3 + 2y^2$, we need to combine the like terms.

1. Identify Like Terms: We have two types of terms in this expression:

   • Terms with $x^3$: $6x^3$ and $-5x^3$
   • Terms with $y^2$: $3y^2$ and $2y^2$

2. Combine $x^3$ Terms: Add the coefficients of the $x^3$ terms:

   • $6x^3 - 5x^3 = (6 - 5)x^3 = 1x^3 = x^3$

3. Combine $y^2$ Terms: Add the coefficients of the $y^2$ terms:

   • $3y^2 + 2y^2 = (3 + 2)y^2 = 5y^2$

4. Combine Simplified Terms: Now, combine the simplified $x^3$ and $y^2$ terms:

   • $x^3 + 5y^2$

Therefore, the equivalent expression is $x^3 + 5y^2$.

Detailed Explanation of Each Step

Step 1: Identifying Like Terms

The first critical step in simplifying any algebraic expression is correctly identifying the like terms. Like terms are terms that have the same variable raised to the same power. In the expression $6x^3 + 3y^2 - 5x^3 + 2y^2$, the terms $6x^3$ and $-5x^3$ are like terms because they both involve the variable $x$ raised to the power of 3. Similarly, $3y^2$ and $2y^2$ are like terms because they both involve the variable $y$ raised to the power of 2. It's crucial to distinguish like terms from unlike terms, such as $x^3$ and $y^2$, which cannot be combined directly because they involve different variables. Misidentifying like terms is a common mistake that can lead to incorrect simplification.

To accurately identify like terms, pay close attention to both the variable and its exponent. For example, $x^3$ and $x^2$ are not like terms because, although they share the same variable, their exponents are different. Similarly, $xy$ and $x$ are not like terms because they do not have the same combination of variables. Once like terms are correctly identified, they can be combined by adding or subtracting their coefficients, which leads us to the next step in the simplification process. This foundational step is essential for successfully simplifying more complex algebraic expressions as well.

Step 2: Combining $x^3$ Terms

Once the like terms have been identified, the next step involves combining them. For the $x^3$ terms in the expression $6x^3 + 3y^2 - 5x^3 + 2y^2$, we have $6x^3$ and $-5x^3$. To combine these terms, we add their coefficients. The coefficient of a term is the numerical factor that multiplies the variable part of the term. In this case, the coefficient of $6x^3$ is 6, and the coefficient of $-5x^3$ is -5. Adding the coefficients gives us $6 + (-5)$, which equals 1.

Therefore, when we combine $6x^3$ and $-5x^3$, we get $1x^3$. In algebraic notation, $1x^3$ is typically written simply as $x^3$, as the coefficient of 1 is often omitted. This simplification is based on the distributive property of multiplication over addition, which states that $ax + bx = (a + b)x$. Applying this property in reverse, we can combine the coefficients of the like terms. The correct application of this step is crucial for simplifying expressions and solving equations. Mistakes in this step often arise from incorrect arithmetic operations with signed numbers or misapplication of the distributive property.

Step 3: Combining $y^2$ Terms

Following the same principle as combining the $x^3$ terms, we now focus on the $y^2$ terms in the expression $6x^3 + 3y^2 - 5x^3 + 2y^2$. We have two terms involving $y^2$: $3y^2$ and $2y^2$. To combine these like terms, we add their coefficients. The coefficient of $3y^2$ is 3, and the coefficient of $2y^2$ is 2. Adding these coefficients gives us $3 + 2$, which equals 5.

Thus, combining $3y^2$ and $2y^2$ results in $5y^2$. This process is analogous to combining the $x^3$ terms and relies on the same distributive property of multiplication over addition. The ability to accurately combine like terms is a foundational skill in algebra and is essential for simplifying more complex expressions and solving equations. Errors in this step are commonly due to simple arithmetic mistakes or a misunderstanding of how to apply the distributive property. Paying close attention to the coefficients and ensuring correct addition will lead to accurate simplification.

Step 4: Combining Simplified Terms

After simplifying the $x^3$ terms and the $y^2$ terms separately, the final step is to combine the simplified terms to obtain the equivalent expression. From Step 2, we found that $6x^3 - 5x^3$ simplifies to $x^3$. From Step 3, we found that $3y^2 + 2y^2$ simplifies to $5y^2$. Now, we combine these results to form the simplified expression.

Combining $x^3$ and $5y^2$ gives us $x^3 + 5y^2$. These terms cannot be combined further because they are not like terms; $x^3$ and $y^2$ involve different variables raised to different powers. Therefore, $x^3 + 5y^2$ is the simplest form of the original expression. This final combination represents the complete simplification of the given expression, and it directly corresponds to one of the answer choices provided in the original problem. Accurately reaching this simplified form demonstrates a solid understanding of algebraic simplification techniques.

Identifying the Correct Option

By following the step-by-step simplification process, we determined that the expression $6x^3 + 3y^2 - 5x^3 + 2y^2$ is equivalent to $x^3 + 5y^2$. Now, we need to match this simplified expression with the options provided in the question.

• A) $6x^3y^2$ - This option is incorrect because it multiplies terms with different variables, which is not what we obtained in our simplification.
• B) $x^6 + 3y^4$ - This option is incorrect because it involves changing the exponents of the variables, which is not allowed when combining like terms.
• C) $x^3 + 5y^2$ - This option matches our simplified expression, so it is the correct answer.
• D) $6x^6y^4$ - This option is incorrect for the same reasons as options A and B; it involves both multiplying unlike terms and changing exponents.

Therefore, the correct answer is C) $x^3 + 5y^2$. This final step confirms that our step-by-step solution has led us to the correct equivalent expression, demonstrating a clear understanding of algebraic simplification.

Conclusion

In conclusion, the expression equivalent to $6x^3 + 3y^2 - 5x^3 + 2y^2$ is $x^3 + 5y^2$, which corresponds to option C. This result was achieved by systematically combining like terms, a fundamental technique in algebraic simplification. By identifying and grouping terms with the same variables and exponents, we were able to reduce the original expression to its simplest form.

The step-by-step solution involved first identifying the like terms: $6x^3$ and $-5x^3$ (terms with $x^3$), and $3y^2$ and $2y^2$ (terms with $y^2$). We then combined the $x^3$ terms by adding their coefficients ($6 - 5 = 1$), resulting in $x^3$. Similarly, we combined the $y^2$ terms by adding their coefficients ($3 + 2 = 5$), resulting in $5y^2$. Finally, we combined the simplified terms $x^3$ and $5y^2$ to obtain the equivalent expression $x^3 + 5y^2$.

This problem highlights the importance of understanding and applying the basic rules of algebra, such as the commutative, associative, and distributive properties. Mastery of these concepts is essential for success in more advanced mathematical topics. By breaking down the problem into manageable steps and carefully executing each step, we were able to arrive at the correct answer with confidence. This process underscores the value of a systematic approach to problem-solving in mathematics.

Final Answer: C) $x^3 + 5y^2$