Solve For N In N-2y=(3y-n)/m When Y=5 And M=-3
Hey guys! Today, we're diving into a fun little algebra problem. We've got the equation n - 2y = (3y - n) / m, and our mission, should we choose to accept it, is to find the value of n when y is 5 and m is -3. Sounds like a plan? Let's jump right in!
Understanding the Problem
Before we start crunching numbers, let's break down what we're dealing with. We have an equation with three variables: n, y, and m. We're given specific values for y and m, and our goal is to figure out what n has to be to make the equation true. Think of it like a puzzle where we have most of the pieces, and we just need to find the missing one. Algebra, at its heart, is all about solving these kinds of puzzles. It's like being a detective, using the clues (the given values and the equation itself) to uncover the mystery (the value of n). So, let's put on our detective hats and get to work!
In this context, the value of n is what we're after, and we're going to use the principles of algebra to isolate n on one side of the equation. This involves a bit of manipulation, like moving terms around and simplifying expressions. Don't worry, it's not as scary as it sounds! We'll take it step by step, and you'll see how each move gets us closer to our goal. It’s important to understand the equation n-2y = (3y-n)/m because this is the foundation of our entire solution. This equation shows the relationship between n, y, and m. If we change any of these values, it will affect the others. That’s why knowing the values of y and m is crucial for us to find n. Think of the equation as a balancing scale. Whatever we do to one side, we have to do to the other to keep it balanced. This principle will guide us as we solve for n.
Step-by-Step Solution
1. Substituting the Values
The first thing we need to do is plug in the values we know. We're told that y = 5 and m = -3. Let's replace those letters with their numerical values in the equation:
n - 2(5) = (3(5) - n) / (-3)
This step is crucial because it transforms our equation from one with three variables to one with just one: n. Now we can focus solely on isolating n. This substitution is like setting the stage for the rest of the solution. It takes us from the abstract (an equation with letters) to the concrete (an equation with numbers and just one unknown). Once we've made this substitution, the rest of the problem becomes a series of algebraic manipulations to get n by itself. So, let's move on to the next step, where we'll start simplifying things!
2. Simplifying the Equation
Now, let's simplify both sides of the equation. First, we'll take care of the multiplication:
n - 10 = (15 - n) / (-3)
Next, to get rid of the fraction, we'll multiply both sides of the equation by -3. Remember, whatever we do to one side, we have to do to the other to keep things balanced:
-3(n - 10) = -3 * [(15 - n) / (-3)]
This simplifies to:
-3n + 30 = 15 - n
This step is all about cleaning up the equation and making it easier to work with. By simplifying, we're reducing the number of operations we need to perform and making the equation more manageable. Getting rid of the fraction is a big win because fractions can often make algebraic manipulations more complicated. Multiplying both sides by -3 effectively cancels out the denominator, leaving us with a cleaner, linear equation. This is a common strategy in algebra: try to eliminate fractions as early as possible to simplify the process. Now that we've simplified the equation, we're in a much better position to isolate n. Let's move on to the next step and get closer to our solution!
3. Isolating n
Our goal now is to get all the n terms on one side of the equation and all the constant terms on the other side. Let's add 3n to both sides:
-3n + 30 + 3n = 15 - n + 3n
This gives us:
30 = 15 + 2n
Now, let's subtract 15 from both sides:
30 - 15 = 15 + 2n - 15
This simplifies to:
15 = 2n
Isolating n is the heart of solving for its value. It's like separating the ingredient we want to measure from all the other ingredients in a recipe. By adding 3n to both sides, we moved all the n terms to the right side of the equation. Then, by subtracting 15 from both sides, we moved all the constant terms to the left side. This process of moving terms around is a fundamental technique in algebra, and it's crucial for solving equations. Each of these steps is carefully chosen to bring us closer to our goal of having n all by itself on one side of the equation. Now that we've isolated n as much as possible, we just have one small step left to find its value. Let's do it!
4. Solving for n
Finally, to solve for n, we'll divide both sides of the equation by 2:
15 / 2 = 2n / 2
This gives us:
n = 15 / 2
Or, in decimal form:
n = 7.5
Dividing both sides by 2 is the final step in our algebraic journey. It's like the last piece of the puzzle fitting into place. By doing this, we've effectively isolated n and found its value. The result, n = 15/2 or 7.5, is the solution to our original equation when y = 5 and m = -3. This is the value of n that makes the equation true. And there you have it! We've successfully solved for n using the principles of algebra. It’s a great feeling when all the steps come together and you arrive at the answer. Now, let’s recap what we’ve done to make sure we’ve got it all straight.
Conclusion
So, after all that algebraic maneuvering, we've found that when y = 5 and m = -3, the value of n in the equation n - 2y = (3y - n) / m is 7.5. Awesome job, guys! We took a seemingly complex problem and broke it down into manageable steps. Remember, algebra is all about understanding the rules and applying them systematically. Keep practicing, and you'll become a pro at solving these kinds of problems in no time. You've got this!
We started by substituting the given values of y and m into the equation. This is a crucial first step because it simplifies the equation and allows us to focus on solving for n. Then, we simplified the equation by multiplying both sides by -3 to eliminate the fraction. This is a common technique in algebra that makes the equation easier to work with. Next, we isolated n by adding 3n to both sides and then subtracting 15 from both sides. This step is all about rearranging the equation so that n is on one side by itself. Finally, we solved for n by dividing both sides by 2, which gave us our answer: n = 7.5. This final step is the culmination of all our hard work, and it’s where we actually find the value of n. Solving this equation is a fantastic example of how algebra can be used to solve real-world problems. It teaches us how to break down complex problems into smaller, more manageable steps and how to use algebraic techniques to find solutions. The principles we’ve used here can be applied to a wide range of problems, both in mathematics and in other fields. So, remember these steps and keep practicing, and you’ll become a master of algebra!