Solve For M: A Step-by-Step Algebraic Solution

Let's dive into solving for 'm' in the equation:

(3/8)m - 2((1/4)m + 3) = (1/4)m

This might look a bit intimidating at first, but don't worry, we'll break it down step by step. We'll go through each stage, giving you not only the answer but also a solid grasp of the underlying math principles. So, grab your pencil and paper, and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we understand what we're dealing with. We have a linear equation with one variable, 'm'. Our goal is to isolate 'm' on one side of the equation to find its value. This involves using various algebraic techniques to simplify the equation and eventually get 'm' by itself. Guys, this is like a puzzle, and we're about to solve it piece by piece!

Breaking Down the Left Side

The left side of the equation, (3/8)m - 2((1/4)m + 3), looks a bit complex, so let's simplify it first. We need to deal with the parentheses and the fractions. The key here is the distributive property, which tells us how to multiply a number by a sum or difference inside parentheses. Remember, it's like sharing the love (or the multiplication, in this case) to everything inside the brackets.

First, let's distribute the -2 across the terms inside the parentheses: 2 * ((1/4)m + 3). This means we multiply -2 by both (1/4)m and 3.

-2 * (1/4)m = - (2/4)m, which simplifies to -(1/2)m. And, -2 * 3 = -6.

So, the left side now becomes: (3/8)m - (1/2)m - 6. See? We're already making progress! It's all about taking it slowly and making sure each step is crystal clear. Now we have something we can work with more easily.

Combining Like Terms

Now that we've handled the parentheses, the next step is to combine the 'm' terms on the left side of the equation. We have (3/8)m and -(1/2)m. To combine these, we need a common denominator. Think of it like adding slices of pizza – you need to make sure the slices are the same size before you can count them up! The smallest common denominator for 8 and 2 is 8. So, let's convert -(1/2)m to an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of -(1/2) by 4: -(1/2) * (4/4) = -(4/8).

Now we have: (3/8)m - (4/8)m - 6. We can now combine the 'm' terms: (3/8)m - (4/8)m = -(1/8)m.

So, the left side of the equation simplifies to: -(1/8)m - 6. We're getting closer to isolating 'm'! Each step we take makes the equation a little simpler, a little easier to manage. This is the power of algebra – breaking down complex problems into smaller, solvable parts.

Isolating 'm'

Now our equation looks like this: -(1/8)m - 6 = (1/4)m. The goal now is to get all the 'm' terms on one side and the constants (the numbers without 'm') on the other side. It's like sorting your laundry – putting all the socks together and all the shirts together. Let's start by getting rid of the -6 on the left side. To do this, we add 6 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This is a fundamental principle of algebra.

Adding 6 to both sides gives us: -(1/8)m - 6 + 6 = (1/4)m + 6. The -6 and +6 on the left side cancel each other out, leaving us with: -(1/8)m = (1/4)m + 6. Now we have all the constants on the right side, but we still have 'm' terms on both sides. Let's get them all on the left!

Moving the 'm' Terms

To move the (1/4)m term from the right side to the left side, we subtract (1/4)m from both sides. Again, we're keeping the equation balanced by doing the same thing to both sides. So, we have: -(1/8)m - (1/4)m = (1/4)m + 6 - (1/4)m. The (1/4)m terms on the right side cancel each other out, leaving us with: -(1/8)m - (1/4)m = 6. Now we just have 'm' terms on the left and a constant on the right.

But we're not quite there yet. We need to combine the 'm' terms on the left. Just like before, we need a common denominator to do this. We have -(1/8)m and -(1/4)m. The common denominator is 8, so we need to convert -(1/4)m to an equivalent fraction with a denominator of 8. We multiply both the numerator and denominator of -(1/4) by 2: -(1/4) * (2/2) = -(2/8).

Now we have: -(1/8)m - (2/8)m = 6. Combining the 'm' terms, we get: -(3/8)m = 6. We're almost there! Just one more step to isolate 'm'.

The Final Step: Solving for 'm'

We have the equation: -(3/8)m = 6. To get 'm' by itself, we need to get rid of the -(3/8) coefficient. We can do this by multiplying both sides of the equation by the reciprocal of -(3/8), which is -(8/3). Remember, the reciprocal of a fraction is just flipping the numerator and denominator. This is the final piece of the puzzle, guys! Multiplying both sides by -(8/3) gives us: -(8/3) * -(3/8)m = 6 * -(8/3). On the left side, the -(8/3) and -(3/8) cancel each other out, leaving us with just 'm'. On the right side, we have 6 * -(8/3), which simplifies to -48/3, and further simplifies to -16.

So, we have: m = -16. We've done it! We've successfully isolated 'm' and found its value. This is the solution to our equation.

Verification

It's always a good idea to check our answer to make sure we haven't made any mistakes. To do this, we substitute m = -16 back into the original equation and see if both sides are equal. This is like checking your work to make sure you've got the right answer. Let's do it!

Original equation: (3/8)m - 2((1/4)m + 3) = (1/4)m

Substitute m = -16: (3/8)(-16) - 2((1/4)(-16) + 3) = (1/4)(-16)

Simplify: -6 - 2(-4 + 3) = -4

Simplify further: -6 - 2(-1) = -4

Simplify: -6 + 2 = -4

Simplify: -4 = -4

Both sides are equal! This confirms that our solution, m = -16, is correct. We nailed it! Verification is a crucial step in problem-solving. It gives you the confidence that your answer is correct, and it can also help you catch any mistakes you might have made along the way.

Conclusion

So, to wrap it up, we've successfully solved for 'm' in the equation (3/8)m - 2((1/4)m + 3) = (1/4)m. We found that m = -16. We did this by carefully simplifying the equation, combining like terms, isolating 'm', and then verifying our solution. This whole process demonstrates the power of algebraic manipulation. Each step built upon the previous one, leading us to the solution.

Remember, guys, math is like building a house. You need to lay a solid foundation (understanding the basic principles) before you can put up the walls and the roof (solve the more complex problems). Don't be afraid to break down complex problems into smaller, more manageable steps. And always, always verify your answers! Keep practicing, and you'll become a math whiz in no time!

This wasn't just about finding the answer; it was about understanding the process. Math isn't just about memorizing formulas; it's about developing a way of thinking, a way of approaching problems logically and systematically. So, take these skills and apply them to other problems. Challenge yourself, and you'll be amazed at what you can achieve.

If you have similar problems, try using this step-by-step approach. Break down the problem, simplify, isolate the variable, and verify your solution. Math is a journey, not a destination. Enjoy the ride, guys, and keep exploring!