Simplifying 5y² - 3y² A Step-by-Step Guide
In mathematics, one of the fundamental skills is the ability to simplify algebraic expressions. This involves combining like terms, which are terms that have the same variable raised to the same power. When you encounter an expression like 5y² - 3y², understanding the correct procedure is crucial for arriving at the right answer. This article will delve into the process of simplifying such expressions, highlighting the common mistakes and providing a clear, step-by-step explanation.
The expression 5y² - 3y² is a classic example of an algebraic expression that can be simplified by combining like terms. The key here is to recognize that both terms contain the variable 'y' raised to the power of 2. This makes them like terms, which means they can be combined through addition or subtraction. The confusion often arises when students are unsure whether to manipulate the coefficients (the numbers in front of the variable) or the exponents (the power to which the variable is raised). The correct approach involves focusing on the coefficients while keeping the variable and its exponent the same. To further clarify, let’s break down the components of this expression and discuss the step-by-step methodology for simplifying it.
Identifying Like Terms
The cornerstone of simplifying algebraic expressions is identifying like terms. Like terms are those that have the same variable raised to the same power. For example, 3x² and 7x² are like terms because they both have the variable 'x' raised to the power of 2. However, 3x² and 7x³ are not like terms because the exponents are different, even though the variable is the same. Similarly, 3x² and 7y² are not like terms because the variables are different, even though the exponents are the same. In the expression 5y² - 3y², both terms contain 'y²', making them like terms and allowing us to combine them. This recognition is the first and most crucial step in simplifying any algebraic expression. Once like terms are identified, the next step involves performing the appropriate arithmetic operation on their coefficients.
The Role of Coefficients
The coefficient is the numerical part of a term that multiplies the variable. In the term 5y², the coefficient is 5, and in the term -3y², the coefficient is -3. When simplifying expressions by combining like terms, the operation (addition or subtraction) is performed on the coefficients only. The variable and its exponent remain unchanged. This is because we are essentially counting how many of a particular term we have. For instance, 5y² can be thought of as five instances of 'y²', and 3y² as three instances of 'y²'. Subtracting 3y² from 5y² means we are taking away three instances of 'y²' from five instances of 'y²', leaving us with two instances of 'y²'. Thus, the coefficients are subtracted (5 - 3), but the 'y²' part stays the same. This concept is vital for avoiding common errors, such as incorrectly adding or subtracting exponents, which will be discussed later in this article.
Analyzing the Given Options
To correctly simplify the expression 5y² - 3y², we must carefully consider the options provided and understand why one is correct while the others are not. The options given highlight common misconceptions about simplifying algebraic expressions. Let’s dissect each option to clarify the correct approach.
Option A: Add the Coefficients (5 + 3)
Option A suggests adding the coefficients, which would mean calculating 5 + 3. This approach is incorrect for subtraction problems. While addition is a valid operation in algebra, it is not the correct one to use in this scenario where the expression involves subtraction. Adding the coefficients would lead to an incorrect simplification of the expression. The correct operation to perform in this case is subtraction, as indicated by the minus sign between the terms. This option reflects a misunderstanding of the basic arithmetic operation required to simplify the given expression. It is crucial to recognize the operation indicated in the expression and apply it correctly to the coefficients.
Option B: Add the Exponents (2 + 2)
Option B proposes adding the exponents, which would mean calculating 2 + 2. This is a fundamental misunderstanding of how exponents work in algebraic simplification. Exponents are only added when multiplying terms with the same base, such as in the expression y² * y², which simplifies to y^(2+2) = y^4. However, when we are adding or subtracting like terms, the exponents remain unchanged. The exponent indicates the power to which the variable is raised and is part of the term's identity. Changing the exponent would change the nature of the term itself. In the expression 5y² - 3y², the exponent 2 signifies that 'y' is squared, and this remains constant throughout the simplification process. Adding exponents in this context would lead to an entirely different and incorrect expression.
Option C: Subtract the Exponents (2 - 2)
Option C suggests subtracting the exponents, which would mean calculating 2 - 2. Like Option B, this is another incorrect application of exponent rules. Subtracting exponents is appropriate when dividing terms with the same base, such as in the expression y^5 / y^2, which simplifies to y^(5-2) = y^3. However, when simplifying by adding or subtracting like terms, the exponents are not altered. The exponents define the degree of the term, and altering them would change the fundamental nature of the term. In the given expression, the exponent 2 is an integral part of the term 'y²', and it must remain unchanged when combining the terms. Subtracting the exponents would lead to a flawed simplification, changing the expression's meaning and value.
Option D: Subtract the Coefficients (5 - 3)
Option D suggests subtracting the coefficients, which means calculating 5 - 3. This is the correct approach to simplifying the expression 5y² - 3y². As discussed earlier, when combining like terms, we perform the indicated arithmetic operation on the coefficients while keeping the variable and its exponent the same. In this case, we subtract the coefficients 5 and 3, which results in 2. Therefore, the simplified expression is 2y². This option correctly identifies the fundamental principle of combining like terms: performing the arithmetic operation on the coefficients while maintaining the integrity of the variable and its exponent. This understanding is crucial for simplifying a wide range of algebraic expressions.
Step-by-Step Solution
To solidify the understanding of the correct approach, let’s walk through a step-by-step solution of the expression 5y² - 3y². This will reinforce the principles discussed and provide a clear methodology for tackling similar problems.
Step 1: Identify Like Terms
The first step is to identify the like terms in the expression. In 5y² - 3y², both terms are like terms because they both contain the variable 'y' raised to the power of 2. Recognizing this is crucial because only like terms can be combined.
Step 2: Focus on the Coefficients
Once like terms are identified, the next step is to focus on the coefficients. The coefficients are the numerical parts of the terms: 5 in 5y² and -3 in -3y². The arithmetic operation between these terms is subtraction, as indicated by the minus sign in the expression.
Step 3: Perform the Subtraction
Now, perform the subtraction operation on the coefficients: 5 - 3 = 2. This calculation tells us how many instances of 'y²' we have after combining the terms. It is important to perform the correct arithmetic operation as indicated in the expression. In this case, subtraction is the correct operation because of the minus sign between the terms.
Step 4: Write the Simplified Expression
Finally, write the simplified expression by combining the result from the coefficient subtraction with the variable and its exponent. The result of subtracting the coefficients is 2, and the variable part is 'y²'. Therefore, the simplified expression is 2y². This step completes the simplification process, providing a clear and concise answer.
The Complete Solution
Putting it all together, the step-by-step solution is:
- Identify like terms: 5y² and -3y²
- Focus on coefficients: 5 and -3
- Perform subtraction: 5 - 3 = 2
- Write the simplified expression: 2y²
This detailed breakdown illustrates the correct methodology for simplifying the expression 5y² - 3y², emphasizing the importance of identifying like terms and performing the appropriate operation on the coefficients.
Common Mistakes to Avoid
Simplifying algebraic expressions can be challenging, and certain mistakes are commonly made. Recognizing these pitfalls can help in avoiding them and ensuring accurate solutions. Here are some common mistakes to watch out for when simplifying expressions like 5y² - 3y²:
Mistake 1: Adding or Subtracting Exponents
One of the most frequent errors is incorrectly adding or subtracting the exponents. As discussed earlier, exponents are only added during multiplication and subtracted during division of terms with the same base. In expressions involving addition or subtraction of like terms, the exponents should not be changed. For instance, in 5y² - 3y², the exponent 2 remains the same throughout the simplification process. Changing the exponent would alter the fundamental nature of the term. This mistake often arises from a misunderstanding of the rules governing exponent manipulation in different algebraic operations.
Mistake 2: Combining Unlike Terms
Another common mistake is attempting to combine unlike terms. Only like terms, which have the same variable raised to the same power, can be combined. For example, 3x² and 7x³ cannot be combined because the exponents are different, and 3x² and 7y² cannot be combined because the variables are different. Trying to combine unlike terms will lead to an incorrect simplification. It's essential to carefully identify like terms before attempting to combine them. This requires a thorough understanding of what constitutes a like term and what does not.
Mistake 3: Incorrectly Applying the Distributive Property
While the distributive property doesn't directly apply to the simple expression 5y² - 3y², it is a common source of error in more complex expressions. The distributive property states that a(b + c) = ab + ac. Applying it incorrectly can lead to significant errors. For example, students might mistakenly try to distribute in situations where it is not appropriate. Understanding when and how to apply the distributive property correctly is crucial for simplifying more complex algebraic expressions.
Mistake 4: Arithmetic Errors with Coefficients
Simple arithmetic errors when adding or subtracting coefficients are also common. For example, incorrectly calculating 5 - 3 as something other than 2 would lead to a wrong answer. These errors can be avoided by carefully performing the arithmetic operations and double-checking the calculations. Even a small mistake in arithmetic can lead to a completely incorrect final answer. Attention to detail in the arithmetic steps is therefore essential.
Mistake 5: Ignoring the Sign of the Coefficient
Failing to account for the sign (positive or negative) of the coefficient is another frequent error. For instance, in the expression 5y² - 3y², the coefficient of the second term is -3. Ignoring the negative sign and simply subtracting 3 would be incorrect. The sign is an integral part of the coefficient and must be considered when performing arithmetic operations. Overlooking the sign can change the entire outcome of the simplification.
Conclusion
In conclusion, the correct statement for solving the expression 5y² - 3y² is D. Subtract the coefficients (5 - 3). This involves identifying the like terms, focusing on their coefficients, and performing the subtraction operation. The simplified expression is 2y². Understanding this process and avoiding common mistakes such as altering exponents or combining unlike terms is crucial for success in algebra. By following a step-by-step approach and paying attention to detail, students can confidently simplify algebraic expressions and build a strong foundation in mathematics. Simplifying expressions is a fundamental skill that underpins more advanced algebraic concepts, making it essential to master the correct methodologies and avoid common pitfalls.
