Solving For P: Making The Equation True

Hey Plastik Magazine readers! Let's dive into a neat little math problem. We're gonna figure out what value of p makes the equation -3p + 1/8 = -1/4 true. Sounds fun, right? Don't worry, it's not as scary as it looks. We'll break it down step by step, making it super easy to follow along. Think of it like a puzzle, and we're the detectives, trying to uncover the mystery number that p represents. This kind of problem is fundamental in algebra, and understanding it will help you with more complex equations down the road. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump into the solution, let's make sure we're all on the same page. This equation is a simple linear equation. The goal is to isolate p on one side of the equation, so we can see what its value is. We achieve this by performing the same operations on both sides of the equation. This maintains the balance, like a perfectly balanced scale. What we do to one side, we must do to the other. Think of it as a golden rule! This helps us keep the equation equivalent throughout the whole process. Now, the equation gives us a starting point. We're given -3p + 1/8 = -1/4. We want to get p by itself. This means we'll need to get rid of everything else that's hanging out with it. We will begin by getting rid of the fraction in the equation. This involves a few simple arithmetic steps. Trust me, it's easier than it might seem at first glance. Remember, the ultimate goal is to find that specific number that, when multiplied by -3, and then having 1/8 added to it, equals -1/4. We are going to isolate p, or p will be all alone on one side of the equation, and the other side will give us its numerical value. So, let's take a look on the next section to see how it works.

Step-by-Step Solution

Alright, let's get down to the nitty-gritty and solve this equation. We'll follow these steps and you will be experts at solving it in no time. First, let's rewrite our equation: -3p + 1/8 = -1/4.

Step 1: Get rid of the fractions

Our first move is to get rid of the fractions. To do this, we'll subtract 1/8 from both sides of the equation. This is key: whatever we do to one side, we must do to the other to keep things balanced. So, our equation becomes: -3p + 1/8 - 1/8 = -1/4 - 1/8. This simplifies to: -3p = -1/4 - 1/8. Now, we need to subtract the fractions on the right side. To do that, we need a common denominator. The least common denominator for 4 and 8 is 8. So, we rewrite -1/4 as -2/8. Our equation now looks like this: -3p = -2/8 - 1/8. Simplifying, we get: -3p = -3/8.

Step 2: Isolate p

Now, we're almost there! We need to isolate p. Currently, p is being multiplied by -3. To undo this, we will divide both sides of the equation by -3. Remember, whatever we do to one side, we must do to the other. So we have: -3p / -3 = (-3/8) / -3. This simplifies to: p = (-3/8) / -3. When dividing a fraction by a whole number, you can think of it as multiplying the denominator by that number. So, (-3/8) / -3 becomes -3 / (8 * -3), which simplifies to -3 / -24.

Step 3: Simplify

Finally, we simplify the fraction. We have p = -3 / -24. A negative divided by a negative is a positive. So, p = 3/24. We can simplify this fraction further by dividing both the numerator and denominator by their greatest common divisor, which is 3. This gives us p = 1/8. So, there you have it, guys! The value of p that makes the equation true is 1/8.

Checking the Solution

It's always a good idea to check your work, right? Let's plug the value of p (which is 1/8) back into the original equation to see if it holds true. Our original equation was -3p + 1/8 = -1/4. We substitute 1/8 for p: -3 * (1/8) + 1/8 = -1/4. This simplifies to: -3/8 + 1/8 = -1/4. Combining the fractions on the left side, we get: -2/8 = -1/4. Finally, simplifying the left side, we have: -1/4 = -1/4. And that's correct! Our left side of the equation equals the right side, meaning our answer is verified. It confirms that our solution is correct. This step is super important to double check our work and avoid any silly mistakes. And it also gives us a confidence boost, knowing we got the right answer. And it also shows that we have a solid understanding of the concepts involved.

Conclusion: The Answer Revealed!

So, to recap, the value of p that makes the equation -3p + 1/8 = -1/4 true is 1/8. We successfully solved for p by isolating it on one side of the equation and performing the same operations on both sides to keep the equation balanced. We got rid of the fractions. We made sure to divide the whole number. This problem demonstrates the fundamental principles of algebra. Congratulations! You've successfully navigated a linear equation. Keep practicing, and you'll become a pro at solving these types of problems. Remember, math is like any other skill. The more you practice, the better you get. Don't be afraid to try different problems, make mistakes (it is part of the learning process), and ask for help when you need it. There are tons of resources available online and in your schools to support your learning journey. So keep up the great work and keep exploring the amazing world of mathematics! Until next time, keep those equations balanced, guys! Feel free to ask any other questions. We are here to help you.