Finding Natural Number M Given Equations 3n-p=4 And 9mn-3mp=648

by Scholario Team 64 views

Hey guys! Today, we're diving into a fascinating mathematical problem where we need to determine the natural number m, given two equations: 3n - p = 4 and 9mn - 3mp = 648. This problem might seem a bit daunting at first, but don't worry, we'll break it down step by step and make it super easy to understand. So, let's put on our thinking caps and get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully grasp what the problem is asking. We're given two equations:

  1. 3n - p = 4
  2. 9mn - 3mp = 648

Our mission is to find the value of the natural number m. Remember, natural numbers are positive whole numbers (1, 2, 3, and so on). The presence of n and p might make it seem complicated, but we'll use some clever algebraic manipulations to isolate m and find its value. Understanding the problem is the first and most crucial step in solving any mathematical challenge.

Breaking Down the Equations

Let's take a closer look at the equations. The first equation, 3n - p = 4, is a simple linear equation with two variables, n and p. The second equation, 9mn - 3mp = 648, looks a bit more complex but notice that it involves the same variables n and p, along with m. This is a key observation! We can likely use the first equation to simplify the second one.

The goal is to manipulate these equations to eliminate either n or p, or both, and get an expression that only involves m. This will allow us to solve for the unknown natural number. It's like a puzzle, and we're piecing together the information to find the solution. Remember, math is all about recognizing patterns and using the right tools to simplify complex problems.

Solving for m: Step-by-Step

Okay, now for the fun part – actually solving for m! Here’s how we’ll do it:

Step 1: Simplify the Second Equation

The second equation, 9mn - 3mp = 648, looks like it can be simplified. Notice that 3m is a common factor on the left side. Let's factor it out:

3m(3n - p) = 648

This is a crucial step! Factoring out the common term makes the equation much easier to handle. It's like decluttering your workspace before starting a big project – it makes everything more manageable.

Step 2: Use the First Equation

Now, remember the first equation? 3n - p = 4. Look familiar? The expression (3n - p) appears in our simplified second equation! This is where the magic happens. We can substitute the value of (3n - p) from the first equation into the second equation:

3m(4) = 648

See how much simpler this looks? This is the power of substitution – replacing a complex expression with its equivalent value to simplify the problem.

Step 3: Isolate m

Now we have a very simple equation: 3m(4) = 648. Let's simplify further:

12m = 648

To isolate m, we divide both sides of the equation by 12:

m = 648 / 12

m = 54

Woohoo! We've found the value of m. This is the moment of triumph when all the hard work pays off!

Step 4: Verify the Solution

It's always a good idea to double-check our answer to make sure it's correct. We found that m = 54. Let's plug this value back into the original equations to see if it works. Since we only needed to find m, and we used the first equation to substitute, we don’t necessarily need to find specific values for n and p to verify that m = 54 is a valid solution given the relationship between the two equations. However, let’s confirm our steps:

We had 3m(3n - p) = 648, and we know 3n - p = 4. Substituting m = 54, we get:

3 * 54 * (4) = 648

162 * 4 = 648

648 = 648

The equation holds true! Verification is the final step to ensure accuracy and build confidence in our solution.

Key Strategies Used

Let's recap the strategies we used to solve this problem. These are valuable techniques that you can apply to other mathematical problems as well:

  1. Simplification: We simplified the second equation by factoring out the common term 3m. This made the equation much easier to work with.
  2. Substitution: We used the first equation to substitute the expression (3n - p) in the second equation. This eliminated the variables n and p and allowed us to solve for m.
  3. Isolation: We isolated the variable m by performing algebraic operations (division) on both sides of the equation.
  4. Verification: We verified our solution by plugging the value of m back into the original equations to ensure it holds true.

These strategies are like tools in your mathematical toolbox. The more you practice using them, the better you'll become at solving complex problems.

Common Mistakes to Avoid

While solving problems like this, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  • Arithmetic Errors: Be careful with your calculations! A simple arithmetic error can throw off the entire solution. Double-check your work, especially when dealing with larger numbers.
  • Incorrect Factoring: Make sure you factor out the correct common term. Double-check that the factored expression is equivalent to the original expression.
  • Misapplication of Substitution: Ensure that you are substituting the correct expression and value. Substituting the wrong thing can lead to incorrect results.
  • Forgetting to Verify: Always verify your solution! This is a crucial step to catch any errors and ensure that your answer is correct.

Learning from mistakes is a key part of the problem-solving process. Don't be afraid to make mistakes – just learn from them and keep practicing!

Practice Problems

Want to test your understanding? Here are a couple of practice problems that are similar to the one we just solved:

  1. Determine the natural number x, knowing that 2a - b = 5 and 4ax - 2bx = 100.
  2. Find the natural number k, given that 5p + q = 12 and 25pk + 5qk = 300.

Working through practice problems is the best way to solidify your understanding and build your problem-solving skills.

Conclusion

And there you have it! We successfully determined the natural number m by simplifying, substituting, and isolating the variable. Remember, the key to solving mathematical problems is to break them down into smaller, manageable steps. By understanding the problem, using the right strategies, and avoiding common mistakes, you can conquer any mathematical challenge that comes your way. Keep practicing, keep learning, and most importantly, keep having fun with math! You've got this!