Solving For M In The Equation (1/2)m - (3/4)n = 16 A Step-by-Step Guide
In mathematics, solving for variables in equations is a fundamental skill. This article will guide you through the process of finding the value of m in the equation (1/2) m - (3/4) n = 16, given that n = 8. We'll break down the steps, explain the underlying concepts, and provide insights that will help you tackle similar problems with confidence. Understanding how to manipulate equations and isolate variables is crucial not only in algebra but also in various real-world applications. This article aims to provide a comprehensive understanding of this process, making it accessible and clear for learners of all levels. Whether you're a student grappling with algebra or someone looking to refresh your math skills, this guide offers a structured approach to solving linear equations. By the end of this article, you'll have a solid grasp of how to substitute values, simplify equations, and ultimately find the solution for m. We will also delve into the importance of each step, ensuring that you not only understand the how but also the why behind each mathematical operation. Let’s embark on this mathematical journey together and unlock the power of algebra! The principles we will explore here are foundational for more advanced mathematical concepts, making this a crucial stepping stone in your mathematical education. Remember, practice is key, and this guide will provide you with the necessary tools to confidently approach similar problems.
1. Understanding the Equation
The equation we're working with is (1/2) m - (3/4) n = 16. This is a linear equation with two variables, m and n. A linear equation is an equation in which the highest power of any variable is 1. The goal is to find the value of m, which means we need to isolate m on one side of the equation. We are given the value of n as 8, which allows us to substitute this value into the equation and simplify it. Understanding the structure of the equation is the first step in solving it. Each term plays a role, and by carefully manipulating these terms, we can unravel the value of the unknown variable. The equation represents a relationship between m and n, and by knowing the value of one variable, we can determine the value of the other. This principle is fundamental in many areas of mathematics and science, where relationships between quantities are often expressed through equations. Before we dive into the steps, let's appreciate the elegance of this equation. It's a concise way to represent a specific relationship, and by applying mathematical principles, we can extract valuable information from it. This is the power of algebra: transforming abstract symbols into concrete solutions. So, let's proceed with confidence, knowing that each step we take brings us closer to unlocking the value of m. Remember, patience and attention to detail are crucial in mathematics. With a clear understanding of the equation, we are well-prepared to tackle the next step: substituting the value of n.
2. Substituting the Value of n
Since we know that n = 8, we can substitute this value into the equation: (1/2) m - (3/4) * (8) = 16. This substitution simplifies the equation by reducing the number of variables. Now, we have an equation with only one variable, m, which makes it easier to solve. The act of substitution is a fundamental technique in algebra. It allows us to replace a variable with its known value, thereby simplifying the equation and bringing us closer to the solution. In this case, substituting n = 8 transforms the equation into a more manageable form. Think of it as replacing a piece in a puzzle; by placing the known value in its correct position, we can see the rest of the puzzle more clearly. The next step involves performing the arithmetic operation resulting from the substitution. We need to multiply (3/4) by 8, which will give us a single numerical term. This simplification is crucial for isolating m and eventually finding its value. Remember, each step in solving an equation is like a step on a journey. We move forward one step at a time, carefully applying the rules of mathematics, until we reach our destination: the solution. So, let's proceed with the next step, confident that we are on the right track. The beauty of mathematics lies in its precision and logic. By following the rules, we can transform complex problems into simple ones, and ultimately find the answers we seek.
3. Simplifying the Equation
After substituting n = 8, our equation becomes (1/2) m - (3/4) * 8 = 16. Now, we need to simplify the term (3/4) * 8. Multiplying 3/4 by 8 gives us 6. So, the equation now looks like this: (1/2) m - 6 = 16. This simplification is a crucial step in isolating m. By performing the multiplication, we've reduced the equation to a simpler form, making it easier to manipulate. Simplification is a core principle in mathematics. It involves reducing complex expressions or equations to their simplest forms, making them easier to understand and work with. In this case, simplifying the term (3/4) * 8 allows us to move closer to isolating m. Think of it as decluttering a room; by removing the unnecessary items, you can see the important things more clearly. The simplified equation, (1/2) m - 6 = 16, is now much easier to work with. We have a single variable, m, and a constant term, -6, on one side of the equation. The next step involves isolating the term containing m by adding 6 to both sides of the equation. This is a fundamental algebraic operation that maintains the equality of the equation while moving us closer to the solution. Remember, each step in solving an equation must be performed on both sides to maintain balance. This is like a seesaw; if you add weight to one side, you must add the same weight to the other side to keep it level. So, let's proceed with the next step, adding 6 to both sides, and continue our journey towards finding the value of m.
4. Isolating the m Term
To isolate the term with m, which is (1/2) m, we need to eliminate the -6 on the left side of the equation. We do this by adding 6 to both sides of the equation: (1/2) m - 6 + 6 = 16 + 6. This simplifies to (1/2) m = 22. Adding the same value to both sides of an equation is a fundamental algebraic principle. It ensures that the equation remains balanced while allowing us to isolate the variable we're solving for. Think of it as balancing a scale; if you add something to one side, you must add the same amount to the other side to maintain equilibrium. In this case, adding 6 to both sides cancels out the -6 on the left side, leaving us with just the term containing m. The equation (1/2) m = 22 is now much closer to our goal. We have isolated the term with m on one side, and a constant value on the other. The next step involves getting rid of the coefficient (1/2) in front of m. We can do this by multiplying both sides of the equation by the reciprocal of 1/2, which is 2. This will effectively isolate m and give us its value. Remember, the goal is to get m by itself on one side of the equation. By carefully applying algebraic principles, we can systematically eliminate the other terms and coefficients until we achieve this goal. So, let's proceed with the next step, multiplying both sides by 2, and bring our journey to a successful conclusion.
5. Solving for m
Now that we have (1/2) m = 22, we need to isolate m. To do this, we multiply both sides of the equation by 2: 2 * (1/2) m = 2 * 22. This simplifies to m = 44. Multiplying both sides of the equation by the reciprocal of the coefficient of m is the final step in isolating m. The reciprocal of 1/2 is 2, so multiplying both sides by 2 effectively cancels out the 1/2 on the left side, leaving us with m by itself. This is the culmination of our efforts – we have successfully solved for m! The solution, m = 44, means that when n is 8, the value of m that satisfies the equation (1/2) m - (3/4) n = 16 is 44. This value can be substituted back into the original equation to verify that it is indeed the correct solution. Verification is a crucial step in problem-solving. It ensures that our solution is accurate and that we have not made any errors along the way. In this case, we can substitute m = 44 and n = 8 into the original equation to see if it holds true. The solution m = 44 represents a specific point on the line defined by the equation (1/2) m - (3/4) n = 16. This line represents all the possible pairs of values for m and n that satisfy the equation. By finding m = 44 when n = 8, we have identified one such pair. We have now successfully navigated the process of solving for m in a linear equation. This process involves understanding the equation, substituting known values, simplifying, isolating the variable, and finally, solving for its value. These skills are fundamental in algebra and will serve you well in more advanced mathematical studies.
6. Verification
To ensure our solution is correct, let's substitute m = 44 and n = 8 back into the original equation: (1/2) * 44 - (3/4) * 8 = 16. Simplifying this, we get 22 - 6 = 16, which is true. Therefore, our solution m = 44 is correct. Verification is a crucial step in problem-solving, especially in mathematics. It provides a check to ensure that our solution satisfies the original conditions of the problem. By substituting the values we found back into the equation, we can confirm whether our solution is accurate. Think of it as proofreading your work; it's a final check to catch any errors before submitting your answer. In this case, the verification process confirms that m = 44 is indeed the correct solution. Substituting m = 44 and n = 8 into the original equation, (1/2) m - (3/4) n = 16, we get: (1/2) * 44 - (3/4) * 8 = 22 - 6 = 16. This confirms that our solution is consistent with the given equation. The verification step not only provides assurance that our solution is correct, but it also reinforces our understanding of the problem and the steps we took to solve it. It's a valuable practice that can help prevent errors and build confidence in our mathematical abilities. Remember, accuracy is paramount in mathematics. By taking the time to verify our solutions, we demonstrate our commitment to precision and attention to detail. So, always make verification a part of your problem-solving routine. It's a small step that can make a big difference.
Conclusion
In conclusion, the value of m in the equation (1/2) m - (3/4) n = 16, when n = 8, is 44. We arrived at this solution by substituting the value of n, simplifying the equation, isolating the m term, and finally solving for m. This process demonstrates the fundamental principles of solving linear equations. Solving equations is a core skill in mathematics and is essential for various applications in science, engineering, and everyday life. The steps we followed in this article – substitution, simplification, isolation, and solving – are applicable to a wide range of algebraic problems. By mastering these techniques, you can confidently tackle more complex equations and mathematical challenges. Remember, practice is key to building proficiency in mathematics. The more you practice solving equations, the more comfortable and confident you will become. Each problem you solve is an opportunity to reinforce your understanding of the underlying principles and to refine your problem-solving skills. So, don't be afraid to tackle new challenges and to seek out opportunities to practice. The journey of learning mathematics is a rewarding one. It requires patience, persistence, and a willingness to learn from mistakes. But the skills and knowledge you gain along the way will empower you to solve problems, think critically, and make informed decisions in all aspects of your life. We hope this guide has been helpful in clarifying the process of solving for variables in linear equations. By following the steps outlined in this article, you can confidently approach similar problems and achieve success in your mathematical endeavors.
