Simplifying Expressions Combining Like Terms In $1.3b + 7.8 - 3.2b$
In mathematics, simplifying expressions is a fundamental skill. One of the key techniques in simplifying expressions is combining like terms. This involves identifying terms that have the same variable raised to the same power and then adding or subtracting their coefficients. This article delves into the process of combining like terms, providing a step-by-step guide with examples to help you master this essential algebraic skill. We will specifically address the expression $1.3b + 7.8 - 3.2b and demonstrate how to simplify it by combining like terms, ensuring a comprehensive understanding of the topic.
Understanding Like Terms
To effectively combine like terms, it is crucial to first understand what constitutes a like term. Like terms are terms that have the same variable raised to the same power. The coefficients, which are the numbers multiplying the variables, can be different. For example, in the expression 5x + 3x - 2, the terms 5x and 3x are like terms because they both have the variable x raised to the power of 1. The term -2 is a constant term, which can also be considered a like term with other constant terms in an expression. Recognizing like terms is the first step in simplifying algebraic expressions.
Consider the expression 4y^2 - 2y + 7y^2 + 5. Here, 4y^2 and 7y^2 are like terms because they both contain the variable y raised to the power of 2. The term -2y is different as it has y raised to the power of 1. The constant term 5 does not have any variable and is thus a like term only with other constant terms, if any exist in the expression. The ability to accurately identify like terms is the foundation for correctly combining them and simplifying the expression.
In more complex expressions, like terms may not be immediately obvious. For instance, in 3ab + 2a - ab + 5b, the terms 3ab and -ab are like terms because they both have the variables a and b multiplied together. The terms 2a and 5b are not like terms as they have different variables. This example highlights the importance of carefully examining each term to identify like terms correctly. A solid grasp of what constitutes like terms is essential for simplifying various algebraic expressions and solving equations effectively.
Identifying Coefficients and Variables
To combine like terms effectively, it's important to clearly identify the coefficients and variables within each term. The coefficient is the numerical part of the term, while the variable is the symbolic part, usually represented by letters like x, y, or b. For instance, in the term 1.3b, the coefficient is 1.3 and the variable is b. Similarly, in the term -3.2b, the coefficient is -3.2 and the variable is b. Recognizing these components is crucial for the next step, which involves adding or subtracting the coefficients of like terms.
Consider the expression 5x^2 + 3x - 2x^2 + 7. Here, the term 5x^2 has a coefficient of 5 and a variable part x^2. The term 3x has a coefficient of 3 and a variable part x. The term -2x^2 has a coefficient of -2 and a variable part x^2. The constant term 7 can be thought of as having a coefficient of 7 and no variable part. By dissecting each term in this way, you can easily identify which terms are like terms and proceed with combining them.
Understanding the role of coefficients and variables is particularly important when dealing with more complex expressions. For example, in the expression 4abc - 2ab + 5abc - 3bc, the terms 4abc and 5abc are like terms because they both have the same variable parts (abc). Their coefficients are 4 and 5, respectively. The terms -2ab and -3bc are not like terms because they have different variable parts. Breaking down each term into its coefficient and variable components allows for a systematic approach to identifying and combining like terms, ensuring accurate simplification of the expression.
Step-by-Step Guide to Combining Like Terms
Combining like terms is a straightforward process that can be broken down into several key steps. By following these steps, you can effectively simplify algebraic expressions and make them easier to work with. The primary goal is to identify terms with the same variable raised to the same power and then combine their coefficients. This systematic approach ensures accuracy and efficiency in algebraic manipulations.
Step 1: Identify Like Terms
The first step in combining like terms is to identify the terms that have the same variable raised to the same power. This involves carefully examining each term in the expression and grouping together those that share the same variable part. For example, in the expression 3x + 2y - 5x + 4y, the like terms are 3x and -5x, as well as 2y and 4y. It's helpful to use visual cues, such as underlining or circling, to group like terms together. This initial step sets the stage for the subsequent steps in the simplification process.
Consider the expression 7a^2 - 3a + 2a^2 + 5a - 1. Here, the like terms are 7a^2 and 2a^2, which both have the variable a raised to the power of 2. The other set of like terms are -3a and 5a, which have the variable a raised to the power of 1. The constant term -1 does not have any like terms in this expression. Correctly identifying like terms is the most critical step, as it dictates which terms can be combined in the following steps.
In more complex expressions, such as 4xy + 2x - 3xy + 5y - x, the process remains the same. The like terms are 4xy and -3xy, which both have the variable parts xy. The terms 2x and -x are also like terms, having the variable x. The term 5y does not have any like terms in this expression. Being meticulous in this initial identification step ensures that you combine only those terms that are mathematically compatible, leading to a simplified and accurate result.
Step 2: Group Like Terms
Once you have identified the like terms, the next step is to group them together. This can be done by rearranging the expression so that like terms are adjacent to each other. Rearranging terms is permissible due to the commutative property of addition, which states that the order in which terms are added does not affect the sum. For example, the expression 3x + 2y - 5x + 4y can be rearranged as 3x - 5x + 2y + 4y. Grouping like terms makes it visually easier to combine their coefficients in the next step.
Consider the expression 7a^2 - 3a + 2a^2 + 5a - 1. After identifying the like terms, you can group them as 7a^2 + 2a^2 - 3a + 5a - 1. This rearrangement clearly shows the like terms together, making the subsequent addition or subtraction of coefficients more straightforward. Proper grouping is a key organizational step that helps prevent errors and ensures accuracy in simplifying expressions.
In more complex scenarios, such as the expression 4xy + 2x - 3xy + 5y - x, grouping like terms involves bringing together terms with the same variable parts. This results in the rearranged expression 4xy - 3xy + 2x - x + 5y. By systematically grouping like terms, you set the stage for combining their coefficients, ultimately leading to a simplified expression. This step is crucial for maintaining clarity and avoiding mistakes when dealing with multiple terms and variables.
Step 3: Combine Coefficients
After grouping the like terms, the final step is to combine their coefficients. This involves adding or subtracting the numerical parts of the like terms while keeping the variable part the same. For example, in the expression 3x - 5x, you would combine the coefficients 3 and -5 to get -2, resulting in the simplified term -2x. Similarly, in the expression 2y + 4y, you would add the coefficients 2 and 4 to get 6, resulting in the term 6y. This step is where the actual simplification occurs, reducing the expression to its most concise form.
Consider the expression 7a^2 + 2a^2 - 3a + 5a - 1. After grouping like terms, you combine the coefficients of a^2 terms (7 + 2 = 9) to get 9a^2. Then, you combine the coefficients of a terms (-3 + 5 = 2) to get 2a. The constant term -1 remains unchanged as there are no other constant terms to combine with. The simplified expression is 9a^2 + 2a - 1. This process demonstrates how combining coefficients reduces the expression to its simplest form.
In the case of the expression 4xy - 3xy + 2x - x + 5y, you combine the coefficients of xy terms (4 - 3 = 1) to get 1xy, which is typically written as xy. For the x terms, you combine the coefficients (2 - 1 = 1) to get 1x, or simply x. The term 5y remains unchanged. The simplified expression is xy + x + 5y. This example illustrates the importance of carefully combining coefficients to achieve an accurate and simplified algebraic expression.
Applying the Steps to the Expression
Now, let's apply the step-by-step guide to the given expression: $1.3b + 7.8 - 3.2b$. This example will provide a practical demonstration of how to combine like terms in a specific algebraic expression. By following the steps outlined earlier, we can simplify this expression efficiently and accurately. The goal is to reduce the expression to its most concise form by identifying and combining the like terms.
Step 1: Identify Like Terms in
The first step is to identify the like terms in the expression $1.3b + 7.8 - 3.2b$. In this expression, we have two terms that contain the variable b: $1.3b and $-3.2b$. These are like terms because they both have the same variable, b, raised to the power of 1. The term $7.8$ is a constant term and does not have any variables. Therefore, the like terms in this expression are $1.3b$ and $-3.2b$. Accurately identifying these terms is crucial for the next steps in the simplification process.
The terms $1.3b$ and $-3.2b$ share the common variable b, making them eligible for combination. The constant term $7.8$ stands alone as it does not have a variable component that matches any other term in the expression. This clear identification of like terms allows us to proceed with grouping them together for the next step. Properly recognizing like terms ensures that only compatible terms are combined, leading to a correct simplification of the expression.
In summary, the key to identifying like terms is to look for terms with the same variable raised to the same power. In the given expression, this means focusing on the terms with b. The constant term is treated separately since it does not have a variable. This initial step is the foundation for simplifying the expression, setting the stage for combining the coefficients in the subsequent steps.
Step 2: Group Like Terms in
Having identified the like terms, the next step is to group them together. In the expression $1.3b + 7.8 - 3.2b$, we rearrange the terms so that the like terms are adjacent to each other. This can be done by moving the $-3.2b$ term next to the $1.3b$ term. The commutative property of addition allows us to rearrange the terms without changing the value of the expression. Therefore, the expression can be rewritten as $1.3b - 3.2b + 7.8$. Grouping like terms makes it visually easier to combine their coefficients in the next step, enhancing clarity and reducing the likelihood of errors.
The rearrangement $1.3b - 3.2b + 7.8$ clearly shows the like terms $1.3b$ and $-3.2b$ positioned next to each other. This proximity makes the subsequent combination of their coefficients more straightforward. The constant term $7.8$ remains at the end of the expression, as it does not have any like terms to group with. Proper grouping is a crucial step in simplifying expressions, as it ensures that like terms are readily available for combination.
By systematically grouping like terms, we are setting the stage for the final step of combining coefficients. This organized approach is essential for handling more complex expressions with multiple sets of like terms. The act of grouping not only simplifies the visual aspect of the expression but also reinforces the understanding of which terms can be combined. This step is a key component of the overall strategy for simplifying algebraic expressions effectively.
Step 3: Combine Coefficients in
Now that we have grouped the like terms, the final step is to combine their coefficients. In the rearranged expression $1.3b - 3.2b + 7.8$, we focus on the terms $1.3b$ and $-3.2b$. To combine these terms, we add their coefficients: $1.3 + (-3.2) = -1.9$. This means that $1.3b - 3.2b$ simplifies to $-1.9b$. The constant term $7.8$ remains unchanged as there are no other constant terms to combine with. Therefore, the simplified expression is $-1.9b + 7.8$. This step completes the simplification process, resulting in a more concise expression.
The process of combining coefficients involves performing the arithmetic operation on the numerical parts of the like terms. In this case, subtracting $3.2$ from $1.3$ yields $-1.9$, which becomes the coefficient of the b term. The resulting term, $-1.9b$, is then combined with the constant term $7.8$. This final combination produces the simplified expression, which is easier to interpret and use in further calculations.
By combining the coefficients, we have reduced the original expression to its simplest form. The expression $-1.9b + 7.8$ is equivalent to the original expression $1.3b + 7.8 - 3.2b$, but it is more streamlined and easier to work with. This final step demonstrates the power of combining like terms in simplifying algebraic expressions, making them more manageable and understandable.
Final Simplified Expression
After applying the steps of identifying, grouping, and combining like terms to the expression $1.3b + 7.8 - 3.2b$, we arrive at the final simplified expression: $-1.9b + 7.8$. This expression is equivalent to the original but is now in its simplest form, with no further like terms to combine. The simplification process has reduced the expression to a more concise and manageable format, making it easier to use in subsequent mathematical operations or problem-solving scenarios. The ability to simplify expressions is a fundamental skill in algebra, and this example demonstrates the practical application of this skill.
The simplified expression $-1.9b + 7.8$ represents the result of combining the like terms $1.3b$ and $-3.2b$ into a single term, $-1.9b$. The constant term $7.8$ remains unchanged as it did not have any like terms to combine with. This final expression is not only simpler but also provides a clearer representation of the relationship between the variable b and the constant term. Simplification is a key technique in mathematics for making expressions more understandable and easier to work with.
In conclusion, the process of combining like terms has transformed the original expression $1.3b + 7.8 - 3.2b$ into its simplified form, $-1.9b + 7.8$. This transformation highlights the importance of understanding and applying algebraic principles to streamline expressions. The simplified expression is the most concise and efficient representation of the original, making it a valuable outcome in various mathematical contexts.
Practice Problems
To further solidify your understanding of combining like terms, it's essential to practice with a variety of problems. Solving practice problems helps reinforce the concepts and techniques discussed, allowing you to apply the steps independently and confidently. Here are a few additional practice problems to help you hone your skills:
- Simplify the expression:
$4x + 2 - 7x + 5$ - Combine like terms in the expression:
$3y^2 - 2y + 5y^2 - y + 4$ - Simplify:
$6a - 3b + 2a + 4b - 1$
Working through these problems will enhance your ability to identify, group, and combine like terms efficiently. Practice is key to mastering algebraic simplification and building a solid foundation in mathematics.
Conclusion
In conclusion, combining like terms is a fundamental algebraic skill that simplifies expressions by grouping and combining terms with the same variable raised to the same power. By following a step-by-step approach—identifying like terms, grouping them together, and combining their coefficients—you can effectively reduce complex expressions to their simplest forms. This article demonstrated how to apply these steps to the expression $1.3b + 7.8 - 3.2b$, resulting in the simplified expression $-1.9b + 7.8$. Mastering this skill is crucial for success in algebra and higher-level mathematics, as it lays the groundwork for solving equations, simplifying more complex expressions, and tackling various mathematical problems. Consistent practice and a clear understanding of the underlying principles will enable you to confidently combine like terms and simplify algebraic expressions.
