Solving For N In -4(n-4)-3=10 A Step-by-Step Guide
Introduction
In this article, we will delve into solving the algebraic equation -4(n-4)-3=10 for the variable n. This type of problem falls under the category of basic algebra, a fundamental topic in mathematics. We will break down the steps involved in isolating n and finding its value. This process includes distribution, combining like terms, and applying inverse operations. Mastering these steps is crucial for success in more advanced mathematical concepts. Our discussion will be clear and comprehensive, ensuring that readers of all levels can follow along and grasp the underlying principles.
Understanding the Equation
Before diving into the solution, it's important to understand the equation we're working with: -4(n-4)-3=10. This equation is a linear equation, meaning the highest power of the variable n is 1. It involves several arithmetic operations, including multiplication, subtraction, and distribution. The goal is to isolate n on one side of the equation to determine its value. This is achieved by systematically applying inverse operations to both sides of the equation, maintaining the balance and equality throughout the process. Linear equations like this are the building blocks of algebra and are used extensively in various fields of science, engineering, and economics.
Step-by-Step Solution
To solve for n in the equation -4(n-4)-3=10, we will proceed step by step:
Step 1: Distribute the -4
The first step is to distribute the -4 across the terms inside the parentheses. This means multiplying -4 by both n and -4. The equation becomes:
-4 * n + (-4) * (-4) - 3 = 10
Simplifying this gives us:
-4n + 16 - 3 = 10
This step is crucial because it removes the parentheses and allows us to combine like terms in the subsequent steps. Distribution is a key algebraic technique and a foundation for simplifying complex expressions.
Step 2: Combine Like Terms
Next, we combine the like terms on the left side of the equation. In this case, the like terms are the constants 16 and -3. Adding these together, we get:
-4n + 13 = 10
Combining like terms simplifies the equation further, making it easier to isolate the variable. This step demonstrates the principle of simplifying equations by grouping similar elements together.
Step 3: Isolate the Term with n
To isolate the term with n, we need to eliminate the constant term on the left side of the equation. In this case, the constant term is 13. We can eliminate it by subtracting 13 from both sides of the equation:
-4n + 13 - 13 = 10 - 13
This simplifies to:
-4n = -3
Isolating the term with the variable is a critical step in solving for n. By applying the same operation to both sides of the equation, we maintain the equality and move closer to finding the value of n.
Step 4: Solve for n
Finally, to solve for n, we need to eliminate the coefficient -4. We can do this by dividing both sides of the equation by -4:
-4n / -4 = -3 / -4
This gives us:
n = 3/4
Therefore, the solution to the equation -4(n-4)-3=10 is n = 3/4. This final step involves using the inverse operation of multiplication (division) to isolate n and determine its value. This demonstrates the core principle of solving equations by undoing the operations performed on the variable.
Verification
To ensure our solution is correct, we can substitute n = 3/4 back into the original equation and verify that it holds true. Let's perform the verification:
Original equation: -4(n-4)-3=10
Substitute n = 3/4:
-4(3/4 - 4) - 3 = 10
First, we simplify the expression inside the parentheses:
3/4 - 4 = 3/4 - 16/4 = -13/4
Now, substitute this back into the equation:
-4(-13/4) - 3 = 10
Multiply -4 by -13/4:
13 - 3 = 10
Finally, subtract 3 from 13:
10 = 10
Since the left side of the equation equals the right side, our solution n = 3/4 is correct. Verification is a vital step in problem-solving, as it confirms the accuracy of the solution and provides confidence in the result. It also reinforces the understanding of the equation and the steps taken to solve it.
Alternative Approaches
While the step-by-step method outlined above is the most common way to solve this type of equation, there are alternative approaches that can be used. One such approach involves working backwards through the order of operations. For instance, instead of distributing the -4 first, one could add 3 to both sides of the equation. This would result in -4(n-4) = 13. Then, one could divide both sides by -4, giving n-4 = -13/4. Finally, adding 4 to both sides would yield n = -13/4 + 4 = -13/4 + 16/4 = 3/4. This alternative method showcases the flexibility in solving algebraic equations and the importance of understanding inverse operations. It also highlights that there may be multiple pathways to arrive at the correct solution.
Another approach could involve recognizing the structure of the equation and using properties of equality to simplify it. For example, one might choose to eliminate the fraction at an earlier stage by multiplying both sides of the equation by 4. While this method might introduce larger numbers into the equation initially, it could simplify the process for some individuals. The key is to choose an approach that aligns with one's understanding and comfort level while ensuring the mathematical principles are followed correctly.
Common Mistakes to Avoid
When solving equations like -4(n-4)-3=10, there are several common mistakes that students often make. Recognizing these pitfalls can help prevent errors and improve accuracy in problem-solving.
Incorrect Distribution
One frequent mistake is the improper distribution of the -4 across the terms inside the parentheses. It's crucial to remember that the -4 must be multiplied by both n and -4. A common error is to only multiply -4 by n, resulting in -4n - 4 instead of the correct -4n + 16. This error can lead to an incorrect solution. To avoid this, always double-check the distribution step and ensure that the negative sign is handled correctly.
Sign Errors
Sign errors are another common source of mistakes. When moving terms across the equals sign, it's essential to change the sign of the term. For example, when adding 3 to both sides of the equation, the -3 on the left side should become +3 on the right side. Forgetting to change the sign or incorrectly changing the sign can lead to an incorrect answer. To mitigate this, it's helpful to write out each step clearly and pay close attention to the signs of the terms.
Order of Operations
Failing to follow the correct order of operations (PEMDAS/BODMAS) can also lead to errors. In this equation, the distribution should be performed before combining like terms. Performing operations in the wrong order can result in an incorrect simplification of the equation. To avoid this, always adhere to the order of operations and ensure that each step is performed in the correct sequence.
Arithmetic Errors
Simple arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can also derail the solution process. Even a small arithmetic mistake can lead to a completely wrong answer. To minimize these errors, it's advisable to double-check all calculations and, if necessary, use a calculator to verify the results.
By being aware of these common mistakes and taking steps to avoid them, students can improve their accuracy and confidence in solving algebraic equations.
Real-World Applications
Solving algebraic equations like -4(n-4)-3=10 isn't just a theoretical exercise; it has numerous real-world applications. Understanding how to manipulate equations to solve for unknowns is a fundamental skill in many fields. Here are a few examples:
Engineering
In engineering, equations are used to model physical systems and solve for unknown variables. For instance, engineers might use equations to calculate the forces acting on a bridge, the flow rate of a fluid through a pipe, or the electrical current in a circuit. The ability to solve equations accurately is crucial for designing safe and efficient systems.
Physics
Physics is a science that relies heavily on mathematical equations to describe the behavior of the universe. Physicists use equations to model everything from the motion of objects to the interactions of subatomic particles. Solving these equations allows them to make predictions about the physical world and test their theories.
Economics
Economists use equations to model economic systems and make predictions about the economy. For example, they might use equations to analyze the relationship between supply and demand, forecast inflation rates, or assess the impact of government policies. The ability to solve equations is essential for understanding and managing complex economic systems.
Computer Science
Computer science relies heavily on algorithms, which are sets of instructions for solving problems. Many algorithms involve mathematical equations, and the ability to solve these equations is crucial for developing efficient and effective software. For instance, equations are used in graphics programming, data analysis, and machine learning.
Everyday Life
Even in everyday life, we encounter situations where solving equations is useful. For example, when planning a budget, calculating a tip at a restaurant, or figuring out the sale price of an item, we are essentially solving equations. The ability to solve equations empowers us to make informed decisions and solve practical problems.
The equation -4(n-4)-3=10 may seem simple, but the skills required to solve it are applicable in a wide range of real-world contexts. By mastering these fundamental algebraic techniques, individuals can unlock a powerful tool for problem-solving in various fields.
Conclusion
In conclusion, solving the equation -4(n-4)-3=10 for n involves a series of algebraic steps, including distribution, combining like terms, isolating the variable term, and dividing to find the value of n. Through this detailed exploration, we have demonstrated that the solution is n = 3/4. We have also emphasized the importance of verifying the solution to ensure its accuracy. Furthermore, we have discussed alternative approaches to solving the equation and highlighted common mistakes to avoid. The real-world applications of solving equations like this are vast, spanning fields such as engineering, physics, economics, computer science, and everyday life. By mastering these fundamental algebraic skills, individuals can enhance their problem-solving abilities and excel in various domains. The principles and techniques discussed in this article provide a solid foundation for tackling more complex algebraic problems in the future.
