Simplifying Algebraic Expressions A Step-by-Step Guide To 2xy + 13 - 3xy - 2

Algebraic expressions form the bedrock of mathematics, and the ability to simplify them is a fundamental skill. Often, these expressions can appear complex at first glance, but with a systematic approach, we can distill them into a more manageable form. In this comprehensive guide, we'll delve into the process of simplifying the algebraic expression $2xy + 13 - 3xy - 2$, breaking down each step and explaining the underlying principles. By the end of this exploration, you'll not only understand how to simplify this specific expression but also gain a broader understanding of algebraic simplification techniques applicable to various problems.

Simplifying algebraic expressions involves combining like terms, which are terms that share the same variables raised to the same powers. Think of it as grouping similar objects together – you can add apples to apples and oranges to oranges, but you can't directly add apples to oranges. In our expression, the terms $2xy$ and $-3xy$ are like terms because they both contain the variables $x$ and $y$ raised to the power of 1. Similarly, the constants $13$ and $-2$ are like terms. The core concept behind simplifying is the distributive property, which allows us to factor out common terms and combine their coefficients. This process reduces the complexity of the expression while maintaining its mathematical equivalence. Mastering this skill is crucial for solving equations, working with functions, and tackling more advanced mathematical concepts. The ability to simplify algebraic expressions not only streamlines calculations but also provides a deeper understanding of the relationships between variables and constants. By meticulously combining like terms and applying the distributive property, we can transform seemingly intricate expressions into their simplest, most elegant forms.

Step 1: Identify Like Terms

The first crucial step in simplifying any algebraic expression is to identify the like terms. Remember, like terms are those that have the same variables raised to the same powers. In the expression $2xy + 13 - 3xy - 2$, we can spot two distinct groups of like terms. First, we have the terms $2xy$ and $-3xy$. Both of these terms contain the variables $x$ and $y$, each raised to the power of 1. This shared variable structure makes them like terms, meaning we can combine them through addition or subtraction. The order of the variables doesn't matter; $xy$ is the same as $yx$. The crucial aspect is that the variables and their exponents are identical.

The second group of like terms in our expression consists of the constants $13$ and $-2$. Constants are numerical values that don't have any variables attached to them. They are always considered like terms because they represent fixed quantities that can be directly added or subtracted. Identifying like terms is like sorting your belongings – you group similar items together before organizing them further. This initial step sets the stage for the simplification process, allowing us to focus on combining the appropriate terms. Without accurately identifying like terms, we risk making errors in the simplification process, leading to an incorrect final expression. Therefore, careful attention to this step is paramount for success. Think of it as the foundation upon which the rest of the simplification process is built. A strong foundation ensures a smooth and accurate simplification.

Step 2: Combine Like Terms

Having successfully identified the like terms, the next step is to combine them. This involves performing the indicated operations (addition or subtraction) on the coefficients of the like terms. Let's first focus on the terms $2xy$ and $-3xy$. These are like terms because they both contain the variables $x$ and $y$ raised to the power of 1. To combine them, we simply add their coefficients: $2 + (-3) = -1$. Therefore, combining $2xy$ and $-3xy$ results in $-1xy$, which is commonly written as $-xy$. It's crucial to pay attention to the signs (positive or negative) of the coefficients when combining terms. A mistake in the sign can lead to an incorrect simplified expression.

Next, we turn our attention to the constant like terms: $13$ and $-2$. These are straightforward to combine as they are simple numerical values. We perform the operation $13 - 2$, which equals $11$. So, the combination of the constant terms yields $11$. Remember, combining like terms is like adding apples to apples – you're simply totaling the quantities of the same type of item. The variable part of the term ($xy$ in this case) remains the same; we only operate on the numerical coefficients. This process is based on the distributive property in reverse, where we're essentially factoring out the common variable part and adding the coefficients. By carefully combining the like terms, we're reducing the number of terms in the expression, making it simpler and easier to understand. This step is the heart of the simplification process, bringing us closer to the final simplified form.

Step 3: Write the Simplified Expression

After meticulously combining the like terms, we arrive at the final step: writing the simplified expression. This involves piecing together the results from the previous step to form the most concise and understandable representation of the original expression. We found that combining the $xy$ terms, $2xy$ and $-3xy$, resulted in $-xy$. Similarly, combining the constant terms, $13$ and $-2$, yielded $11$. Now, we simply put these two results together. The simplified expression is the sum of these two combined terms: $-xy + 11$. This expression is mathematically equivalent to the original expression, $2xy + 13 - 3xy - 2$, but it is much simpler and easier to work with.

The simplified expression contains only two terms, whereas the original expression had four. This reduction in the number of terms makes the expression more manageable and easier to interpret. The order in which you write the terms in the simplified expression generally doesn't matter, as addition is commutative (i.e., $a + b = b + a$). However, it is conventional to write the term with the variable first, followed by the constant term, as we have done here. The simplified expression, $-xy + 11$, represents the most basic form of the original expression. There are no more like terms to combine, and the expression is as concise as it can be. This final step completes the simplification process, providing us with a clear and efficient representation of the initial algebraic expression. This simplified form is not only easier to use in further calculations but also provides a clearer understanding of the relationship between the variables and constants involved.

Final Answer

Therefore, after simplifying the algebraic expression $2xy + 13 - 3xy - 2$, we arrive at the final answer: $-xy + 11$.

This concise expression represents the simplified form of the original expression, achieved by meticulously identifying and combining like terms. The process involved grouping terms with the same variables and exponents, and then performing the necessary arithmetic operations on their coefficients. The constants were similarly combined to yield a single constant term. The resulting expression, $-xy + 11$, is mathematically equivalent to the original but presents it in a more streamlined and easily understandable manner. This simplification not only reduces the number of terms but also clarifies the relationship between the variables and constants within the expression. Mastering this skill of simplifying algebraic expressions is crucial for success in higher-level mathematics, as it forms the foundation for solving equations, working with functions, and tackling more complex algebraic problems. The ability to manipulate and simplify expressions efficiently allows for a deeper understanding of mathematical concepts and facilitates problem-solving in various contexts.