Identifying Like Terms In The Expression 7xy - 9x^2y - 15xy^2 - 14xy
In algebra, identifying like terms is a fundamental skill for simplifying expressions and solving equations. Like terms are terms that have the same variables raised to the same powers. Only like terms can be combined by adding or subtracting their coefficients. This article will delve into the concept of like terms, explain how to identify them, and provide examples to illustrate the process. We will specifically address the expression $7xy - 9x^2y - 15xy^2 - 14xy$ and determine which terms are like terms.
Understanding Like Terms
To effectively identify like terms, it's crucial to understand the definition thoroughly. Like terms are terms that share the same variables, and each variable must be raised to the same power. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical for terms to be considered 'like'. For instance, $3x^2$ and $-5x^2$ are like terms because they both have the variable x raised to the power of 2. However, $3x^2$ and $3x$ are not like terms because the powers of x are different. Similarly, $2xy$ and $7xy$ are like terms because they both contain the variables x and y, each raised to the power of 1. On the other hand, $2xy$ and $2x^2y$ are not like terms because the variable x has different powers in the two terms.
When identifying like terms, pay close attention to the order of the variables and their exponents. For example, $xy$ and $yx$ are like terms because multiplication is commutative, meaning the order of variables does not affect the term's identity. However, $xy$ and $x^2y$ are not like terms due to the different exponents of x. A thorough understanding of these nuances is essential for accurately identifying and combining like terms, which is a critical step in simplifying algebraic expressions and solving equations. Recognizing like terms allows us to consolidate expressions, making them easier to work with and understand. This process is not just a mechanical task; it builds a deeper understanding of the structure of algebraic expressions and the relationships between their components. As you gain proficiency in identifying like terms, you will find that your ability to manipulate algebraic expressions and solve equations significantly improves.
Key Components of Like Terms
To break it down further, let's examine the key components that define like terms:
- Variables: Like terms must contain the same variables. For example, terms with x and y can only be like terms with other terms that also have x and y.
- Exponents: The corresponding variables in like terms must have the same exponents. For instance, $x^2$ and $x^3$ are different, so terms containing these variables raised to these powers are not like terms.
- Coefficients: The coefficients (the numerical part of the term) do not need to be the same. For example, $5x$ and $-3x$ are like terms even though the coefficients are 5 and -3, respectively.
By focusing on these components, one can systematically identify like terms in any algebraic expression. This skill is the bedrock of simplifying expressions, combining terms, and solving equations effectively.
Analyzing the Expression: $7xy - 9x^2y - 15xy^2 - 14xy$
Now, let's apply our understanding of like terms to the expression $7xy - 9x^2y - 15xy^2 - 14xy$. Our goal is to identify which terms in this expression can be combined because they share the same variable factors with the same exponents. We will systematically examine each term and compare it with the others to determine if they meet the criteria for being like terms. This careful analysis will allow us to simplify the expression by combining these like terms, which is a fundamental step in solving algebraic problems.
To begin, we need to break down each term into its components, paying close attention to the variables and their exponents. The first term, $7xy$, has x and y each raised to the power of 1. The second term, $-9x^2y$, has x raised to the power of 2 and y raised to the power of 1. The third term, $-15xy^2$, has x raised to the power of 1 and y raised to the power of 2. Finally, the fourth term, $-14xy$, has both x and y raised to the power of 1. By dissecting each term in this way, we can more easily compare them and identify those that are like terms. Remember, the key is to match not just the variables but also their exponents. This meticulous approach is crucial for ensuring that we combine only terms that are truly alike, which maintains the integrity of the expression and leads to correct simplification.
Step-by-Step Identification
Let's examine each term in the expression $7xy - 9x^2y - 15xy^2 - 14xy$:
- First term: $7xy$ (x and y, both to the power of 1)
- Second term: $-9x^2y$ (x to the power of 2, y to the power of 1)
- Third term: $-15xy^2$ (x to the power of 1, y to the power of 2)
- Fourth term: $-14xy$ (x and y, both to the power of 1)
By carefully comparing these terms, we can pinpoint which ones share the same variable factors with identical exponents. This meticulous comparison is crucial for accurately identifying like terms and avoiding the mistake of combining terms that are not alike.
Identifying Like Terms in the Expression
After analyzing the expression $7xy - 9x^2y - 15xy^2 - 14xy$, we can now identify the like terms. Remember, like terms have the same variables raised to the same powers. Comparing the terms:
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7xy$ has *x* and *y*, both to the power of 1.
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-9x^2y$ has *x* to the power of 2 and *y* to the power of 1.
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-15xy^2$ has *x* to the power of 1 and *y* to the power of 2.
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-14xy$ has *x* and *y*, both to the power of 1.
From this comparison, it's clear that $7xy$ and $-14xy$ are like terms because they both have x and y raised to the power of 1. The other terms, $-9x^2y$ and $-15xy^2$, are not like terms with $7xy$ and $-14xy$ because they have different powers for x and y.
This identification process is a key step in simplifying algebraic expressions. By recognizing and grouping like terms, we can consolidate the expression into a more manageable form. In this case, identifying $7xy$ and $-14xy$ as like terms allows us to combine them, which is a fundamental operation in algebra. The ability to accurately identify like terms is crucial for mastering algebraic manipulations and solving equations effectively.
Combining Like Terms
Having identified $7xy$ and $-14xy$ as like terms in the expression $7xy - 9x^2y - 15xy^2 - 14xy$, we can now combine them. To combine like terms, we simply add or subtract their coefficients while keeping the variable part the same. In this case, we have:
The coefficients are 7 and -14. Adding these coefficients gives us:
Therefore, combining the like terms $7xy$ and $-14xy$ results in $-7xy$. The other terms in the original expression, $-9x^2y$ and $-15xy^2$, are not like terms with $-7xy$ and cannot be combined with it. They remain as they are in the simplified expression.
The process of combining like terms is a fundamental operation in algebra that simplifies expressions and makes them easier to work with. By consolidating like terms, we reduce the complexity of the expression, making it more straightforward to analyze and manipulate. This simplification is not just a cosmetic change; it often reveals the underlying structure of the expression and can be a crucial step in solving equations or performing further algebraic operations.
Final Simplified Expression
After combining the like terms in the expression $7xy - 9x^2y - 15xy^2 - 14xy$, we arrive at the simplified expression:
This simplified expression is equivalent to the original expression, but it contains fewer terms, making it easier to understand and work with. The terms $-7xy$, $-9x^2y$, and $-15xy^2$ are not like terms, so they cannot be combined further. Each term has a unique combination of variables and exponents, preventing any further simplification by combining terms.
The process of simplifying algebraic expressions by identifying and combining like terms is a cornerstone of algebra. It allows us to reduce complex expressions to their simplest forms, making them more manageable for further operations, such as solving equations or evaluating expressions for specific values of the variables. The ability to confidently simplify expressions is a crucial skill for success in algebra and beyond.
Conclusion
In conclusion, when given the expression $7xy - 9x^2y - 15xy^2 - 14xy$, the like terms are $7xy$ and $-14xy$. Combining these like terms simplifies the expression to $-7xy - 9x^2y - 15xy^2$. Understanding how to identify and combine like terms is essential for simplifying algebraic expressions and solving equations effectively. This skill forms a foundation for more advanced algebraic concepts and is crucial for success in mathematics.
