Simplifying Algebraic Expressions A Step-by-Step Guide To $7(6m+5)-(8-5m)-7$

Introduction to Simplifying Algebraic Expressions

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to manipulate complex equations into more manageable forms, making them easier to understand and solve. This article will delve into the process of simplifying the expression $7(6m+5)-(8-5m)-7$, providing a step-by-step guide that enhances comprehension and mathematical proficiency. Before we begin, it's crucial to understand the basic principles governing algebraic manipulation. These principles include the distributive property, combining like terms, and the order of operations. The distributive property states that $a(b+c) = ab + ac$, which is essential for expanding expressions. Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. Mastering these concepts is key to successfully simplifying any algebraic expression. With a firm grasp of these foundational principles, we can approach the given expression with confidence, breaking it down into smaller, more digestible parts. This methodical approach not only simplifies the expression but also reinforces our understanding of algebraic principles. Let's embark on this journey of simplification, transforming the complex into the clear and concise.

Step 1: Applying the Distributive Property

Our initial task in simplifying the expression $7(6m+5)-(8-5m)-7$ is to apply the distributive property. This property is a cornerstone of algebraic manipulation, allowing us to eliminate parentheses and make the expression more manageable. The distributive property, mathematically expressed as $a(b+c) = ab + ac$, dictates that we multiply the term outside the parentheses by each term inside the parentheses. In our expression, the term $7(6m+5)$ calls for the application of this property. We multiply $7$ by both $6m$ and $5$, which yields $7 * 6m + 7 * 5$. This simplifies to $42m + 35$. By applying the distributive property, we've successfully expanded the first part of our expression, removing the parentheses and paving the way for further simplification. This step is crucial because it transforms a more complex term into a sum of simpler terms, making it easier to combine like terms later on. The ability to correctly apply the distributive property is a fundamental skill in algebra, and mastering it is essential for tackling more complex expressions and equations. Now that we've handled the first set of parentheses, let's move on to the next part of the expression and continue our simplification journey. Remember, each step we take brings us closer to the final, simplified form.

Step 2: Dealing with the Negative Sign

Having applied the distributive property to the first part of our expression, we now turn our attention to the term $-(8-5m)$. This step is crucial because the negative sign in front of the parentheses requires careful handling. The negative sign can be thought of as multiplying the entire expression inside the parentheses by $-1$. This means we need to distribute the $-1$ to both terms inside the parentheses, similar to how we applied the distributive property in the previous step. So, $-(8-5m)$ becomes $-1 * 8 + (-1) * (-5m)$. This simplifies to $-8 + 5m$. It's essential to pay close attention to the signs when distributing the negative sign, as an incorrect sign can lead to an incorrect final answer. The negative sign effectively changes the sign of each term inside the parentheses: the positive $8$ becomes negative, and the negative $5m$ becomes positive. This step highlights the importance of precision in algebraic manipulation. By correctly handling the negative sign, we ensure that our expression remains mathematically accurate as we continue the simplification process. Now that we've addressed this part of the expression, we can move on to combining like terms, which will bring us closer to our final simplified form. Each step we take, with careful attention to detail, contributes to a clear and concise result.

Step 3: Combining Like Terms

After applying the distributive property and dealing with the negative sign, our expression now looks like this: $42m + 35 - 8 + 5m - 7$. The next crucial step in simplifying this expression is combining like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two types of terms: terms with the variable $m$ and constant terms (numbers without a variable). To combine like terms, we simply add or subtract the coefficients (the numbers in front of the variables) of the terms with the same variable and add or subtract the constant terms. Let's start by identifying the terms with the variable $m$: $42m$ and $5m$. Adding these terms together gives us $42m + 5m = 47m$. Next, we identify the constant terms: $35$, $-8$, and $-7$. Combining these terms gives us $35 - 8 - 7 = 20$. By combining like terms, we've significantly reduced the complexity of our expression. We've grouped together the terms that are similar, making the expression more concise and easier to understand. This step is a fundamental aspect of algebraic simplification, and mastering it is crucial for solving more complex equations and problems. Now that we've combined like terms, we have a simplified expression that represents the original expression in its most basic form. This simplified form is not only easier to work with but also provides a clearer understanding of the relationship between the variables and constants in the expression.

Step 4: The Final Simplified Expression

Having meticulously applied the distributive property, managed the negative sign, and combined like terms, we arrive at the final simplified expression. From our previous steps, we combined the terms with the variable $m$ to get $47m$ and the constant terms to get $20$. Therefore, the simplified form of the original expression $7(6m+5)-(8-5m)-7$ is $47m + 20$. This final expression is significantly more concise and easier to understand than the original. It clearly shows the relationship between the variable $m$ and the constant term. The process of simplification we've undertaken demonstrates the power of algebraic manipulation. By applying fundamental principles and following a systematic approach, we've transformed a complex expression into a simple, elegant form. This ability to simplify expressions is not just a mathematical exercise; it's a crucial skill that has wide-ranging applications in various fields, including physics, engineering, and computer science. The final simplified expression, $47m + 20$, is not only the answer to our specific problem but also a testament to the effectiveness of algebraic techniques. It represents the culmination of our step-by-step journey, where each step built upon the previous one to arrive at a clear and concise result. This process reinforces the importance of precision, attention to detail, and a solid understanding of mathematical principles in achieving successful simplification.

Conclusion: The Significance of Algebraic Simplification

In conclusion, the journey of simplifying the algebraic expression $7(6m+5)-(8-5m)-7$ has been a valuable exercise in applying fundamental mathematical principles. We've seen how the distributive property, careful handling of negative signs, and the combination of like terms can transform a complex expression into a simple, elegant form: $47m + 20$. This process not only demonstrates the power of algebraic manipulation but also highlights the importance of precision and attention to detail in mathematics. Algebraic simplification is more than just a mathematical technique; it's a critical skill that underpins many areas of science, technology, engineering, and mathematics (STEM). It allows us to model real-world problems mathematically, analyze them, and find solutions. From physics equations describing motion to economic models predicting market behavior, the ability to simplify expressions is essential for making sense of the world around us. Moreover, the process of simplification cultivates problem-solving skills that are transferable to various domains. It teaches us to break down complex problems into smaller, manageable parts, identify patterns, and apply logical reasoning to arrive at a solution. These skills are invaluable not only in academic pursuits but also in professional and personal life. As we've seen in this article, simplifying expressions requires a systematic approach, a deep understanding of mathematical principles, and the ability to apply these principles with precision. By mastering these skills, we empower ourselves to tackle more complex mathematical challenges and to use mathematics as a powerful tool for understanding and solving problems in the world. The final simplified expression, $47m + 20$, stands as a testament to the effectiveness of these skills and the beauty of mathematical simplification.