Finding The Value Of M In The Equation (1/2)m - (3/4)n = 16 When N = 8
In the realm of algebra, solving equations is a fundamental skill. This article delves into a specific algebraic problem, providing a step-by-step solution and a comprehensive explanation to enhance your understanding. We will tackle the equation 1/2 m - 3/4 n = 16, where we aim to determine the value of m when n is given as 8. This problem exemplifies the core principles of solving linear equations, which are essential for various mathematical and scientific applications. Understanding the process not only helps in solving similar problems but also builds a strong foundation for more advanced algebraic concepts. Let's embark on this mathematical journey to unlock the value of m.
Introduction to the Problem
The problem at hand involves a linear equation with two variables, m and n. Our goal is to find the value of m when n is given a specific value. This type of problem is common in algebra and requires a clear understanding of equation manipulation. The given equation is:
1/2 *m* - 3/4 *n* = 16
We are also given that n = 8. To solve for m, we will substitute the value of n into the equation and then isolate m using algebraic operations. This process involves adding or subtracting terms from both sides of the equation and multiplying or dividing both sides by a constant. The key is to maintain the equality of the equation throughout the process. By carefully following these steps, we can determine the unique value of m that satisfies the equation when n is 8. This exercise not only reinforces the practical application of algebraic principles but also sharpens problem-solving skills.
Step-by-Step Solution
To determine the value of m in the equation, we will proceed with a step-by-step approach. First, we substitute the given value of n into the equation:
1/2 *m* - 3/4 * (8) = 16
Next, we simplify the equation by performing the multiplication:
1/2 *m* - 6 = 16
Now, to isolate the term with m, we add 6 to both sides of the equation:
1/2 *m* - 6 + 6 = 16 + 6
This simplifies to:
1/2 *m* = 22
Finally, to solve for m, we multiply both sides of the equation by 2:
2 * (1/2 *m*) = 2 * 22
This gives us the value of m:
m = 44
Therefore, the value of m in the equation 1/2 m - 3/4 n = 16 when n = 8 is 44. This methodical approach demonstrates how algebraic equations can be solved by systematically isolating the variable of interest. Understanding each step ensures accuracy and builds confidence in solving similar problems.
Verification of the Solution
To ensure the accuracy of our solution, we can verify the value of m by substituting it back into the original equation along with the given value of n. The original equation is:
1/2 *m* - 3/4 *n* = 16
We found that m = 44 when n = 8. Substituting these values into the equation, we get:
1/2 * (44) - 3/4 * (8) = 16
Now, we simplify the equation:
22 - 6 = 16
16 = 16
Since the equation holds true, our solution is correct. This verification step is crucial in mathematical problem-solving as it confirms that the calculated value satisfies the given conditions. By substituting the values back into the original equation, we eliminate potential errors and ensure the reliability of our answer. This practice reinforces the importance of precision and accuracy in algebraic manipulations.
Alternative Approaches to Solving the Equation
While the step-by-step method provides a clear and direct solution, exploring alternative approaches can deepen our understanding of algebraic problem-solving. One such approach involves manipulating the equation to eliminate fractions before substituting the value of n.
Starting with the original equation:
1/2 *m* - 3/4 *n* = 16
We can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 4. This will eliminate the fractions:
4 * (1/2 *m* - 3/4 *n*) = 4 * 16
Distributing the 4, we get:
2*m* - 3*n* = 64
Now, we substitute n = 8 into the equation:
2*m* - 3 * (8) = 64
Simplifying:
2*m* - 24 = 64
Adding 24 to both sides:
2*m* = 88
Finally, dividing both sides by 2:
m = 44
This alternative approach yields the same result, m = 44. Demonstrating the flexibility of algebraic methods, this approach highlights how manipulating equations strategically can simplify the solving process. By understanding different methods, we can choose the most efficient one for a given problem.
Common Mistakes and How to Avoid Them
When solving algebraic equations, it's crucial to be aware of common mistakes to ensure accurate results. One frequent error is incorrect arithmetic, such as miscalculating when simplifying terms or performing operations. To avoid this, double-check each step and use a calculator if necessary. Another common mistake is failing to apply the same operation to both sides of the equation, which violates the fundamental principle of equality. Always remember that any operation performed on one side must be mirrored on the other side to maintain balance.
Another pitfall is mishandling negative signs. Pay close attention to the signs of the terms when adding, subtracting, multiplying, or dividing. A simple sign error can lead to a completely incorrect answer. Additionally, students sometimes forget the order of operations (PEMDAS/BODMAS), leading to incorrect simplifications. Remember to address parentheses/brackets first, then exponents/orders, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right).
Finally, failing to verify the solution is a significant oversight. Always substitute the calculated value back into the original equation to check for accuracy. By being mindful of these common mistakes and adopting careful, methodical problem-solving techniques, you can significantly improve your accuracy and confidence in algebra.
Real-World Applications of Solving Linear Equations
Solving linear equations isn't just an abstract mathematical exercise; it has numerous real-world applications. These equations are fundamental tools in various fields, including science, engineering, economics, and finance. In physics, for example, linear equations are used to describe motion, calculate forces, and analyze circuits. Engineers rely on them for designing structures, optimizing systems, and controlling processes.
In economics and finance, linear equations help model supply and demand, calculate interest rates, and analyze investments. Business professionals use them for budgeting, forecasting, and cost analysis. For instance, determining the break-even point for a product or service involves solving a linear equation. Similarly, calculating the monthly payments on a loan or mortgage utilizes linear equations.
Linear equations also play a crucial role in everyday problem-solving. From calculating proportions in recipes to determining travel time based on speed and distance, these equations provide a framework for making informed decisions. Understanding how to solve linear equations empowers individuals to analyze situations quantitatively and find practical solutions. The versatility of linear equations makes them an indispensable tool in a wide range of contexts, highlighting the importance of mastering this fundamental algebraic skill.
Conclusion
In summary, we have successfully determined the value of m in the equation 1/2 m - 3/4 n = 16 when n = 8. Through a step-by-step solution, we found that m = 44. We also verified this solution by substituting the values back into the original equation, confirming its accuracy. Furthermore, we explored an alternative approach to solving the equation, which involved eliminating fractions to simplify the process. This demonstrated the flexibility and adaptability of algebraic methods.
We also discussed common mistakes to avoid when solving equations, emphasizing the importance of careful arithmetic, attention to signs, adherence to the order of operations, and verification of the solution. Finally, we highlighted the real-world applications of solving linear equations, illustrating their relevance in various fields such as science, engineering, economics, and everyday problem-solving. Mastering the techniques for solving linear equations is a valuable skill that enhances analytical and problem-solving abilities, making it an essential component of mathematical literacy.
