Solving For P: Linear Equation 24p + 12 - 18p = 10 + 2p - 6

Hey guys! Today, we're diving into a bit of algebra to figure out the value of p in the linear equation 24p + 12 - 18p = 10 + 2p - 6. Don't worry if equations make you sweat a little; we're going to break it down step by step so it's super easy to follow. Let's get started and make math a little less mysterious, shall we?

Understanding Linear Equations

Before we jump into solving for p, let's quickly recap what a linear equation actually is. Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because when you graph them, they form a straight line. The variable in our equation, in this case p, represents an unknown value that we're trying to find.

Think of a linear equation like a balancing act. On one side of the equals sign, we have some mathematical expressions, and on the other side, we have more expressions. Our goal is to manipulate these expressions, keeping the equation balanced, until we isolate p on one side. When p is all alone on one side, we'll know its value.

Why are linear equations so important? Well, they're everywhere! They pop up in science, economics, computer science – you name it. They help us model and solve problems in all sorts of real-world situations. For example, you might use a linear equation to calculate how long it will take to drive a certain distance at a constant speed, or to figure out how many items you need to sell to break even in your new business venture.

So, with that in mind, let’s look at our equation again: 24p + 12 - 18p = 10 + 2p - 6. See how it fits the definition? We have terms with p (like 24p and -18p) and constant terms (like 12, 10, and -6). Our mission, should we choose to accept it, is to tidy up this equation and find out exactly what value of p makes it true. Ready to roll up our sleeves and get algebraic?

Step 1: Simplify Both Sides of the Equation

The first thing we want to do when tackling any equation, especially a linear one like 24p + 12 - 18p = 10 + 2p - 6, is to make it look a little less intimidating. Think of it as decluttering your room before you start organizing – it just makes everything easier to handle. In math terms, this means simplifying both sides of the equation by combining like terms.

On the left side, we've got 24p and -18p, both of which are terms involving p. We can combine them just like we'd combine apples and oranges (well, almost!). If we have 24 p's and we take away 18 p's, we're left with 6 p's. So, 24p - 18p simplifies to 6p. We also have the constant term +12 hanging out on the left side, so for now, we'll just leave it as is.

That means the left side of our equation now looks like 6p + 12. Much cleaner, right?

Now let's turn our attention to the right side of the equation: 10 + 2p - 6. Here, we have a constant term 10 and another constant term -6. Just like before, we can combine these. If we start with 10 and subtract 6, we get 4. So, 10 - 6 simplifies to 4. We also have the term 2p on the right side, which we'll leave as is for now.

So, the right side of our equation simplifies to 2p + 4. Now, our entire equation looks like this: 6p + 12 = 2p + 4. See how much simpler that is? By combining like terms, we've transformed our original equation into something much more manageable. This step is crucial because it sets us up for the next stage, where we'll start isolating p. Stay with me, guys – we're making progress!

Step 2: Isolate the Variable Term

Okay, now that we've simplified both sides of our equation, 6p + 12 = 2p + 4, it's time to get serious about isolating p. Remember, our goal is to get p all by itself on one side of the equation so we can see its value. To do this, we need to move all the terms with p to one side and all the constant terms to the other side. It's like sorting your socks – you want all the pairs together, right?

Let's start by moving the p terms. We have 6p on the left side and 2p on the right side. A good strategy is to move the smaller p term to the side with the larger p term. This helps us avoid dealing with negative coefficients later on. So, we'll move the 2p from the right side to the left side.

How do we move a term from one side of the equation to the other? We use the magic of inverse operations! Since 2p is being added on the right side, we'll subtract 2p from both sides of the equation. Remember, we have to do the same thing to both sides to keep the equation balanced. If we subtract 2p from both sides, our equation becomes:

6p + 12 - 2p = 2p + 4 - 2p

Now, let's simplify. On the left side, 6p - 2p gives us 4p, so we have 4p + 12. On the right side, 2p - 2p cancels out (that's the whole point!), leaving us with just 4. So, our equation is now:

4p + 12 = 4

We've successfully moved all the p terms to the left side. High five! Now, we need to move the constant terms to the right side. We have +12 on the left side, so we'll do the opposite – we'll subtract 12 from both sides. This gives us:

4p + 12 - 12 = 4 - 12

Simplifying again, 12 - 12 cancels out on the left side, leaving us with just 4p. On the right side, 4 - 12 is -8. So, our equation is now:

4p = -8

Look at that! We've isolated the variable term. We're one step closer to finding p. You guys are doing awesome!

Step 3: Solve for p

Alright, we've reached the final stretch! We've simplified our equation and isolated the variable term, so we're now looking at 4p = -8. Remember, our mission is to get p all by itself so we can see its value. Right now, p is being multiplied by 4. So, what's the opposite of multiplying by 4? Dividing by 4, of course!

To get p by itself, we'll divide both sides of the equation by 4. This is crucial – whatever we do to one side, we have to do to the other to keep things balanced. So, we have:

(4p) / 4 = (-8) / 4

On the left side, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with just p. On the right side, -8 divided by 4 is -2. So, our equation simplifies to:

p = -2

Boom! We did it! We've found the value of p. It turns out that p is equal to -2. Give yourselves a pat on the back – you've successfully navigated this algebraic adventure.

Step 4: Verify the Solution

Before we declare victory and move on, there's one important step we should always take: verifying our solution. Think of it as double-checking your work to make sure you haven't made any sneaky mistakes along the way. It's like proofreading a paper or testing a recipe – it's always good to make sure everything works as expected.

To verify our solution, we'll plug the value we found for p (which is -2) back into our original equation: 24p + 12 - 18p = 10 + 2p - 6. If our solution is correct, then both sides of the equation should be equal when we substitute -2 for p.

Let's start with the left side. We have 24p + 12 - 18p. Replacing p with -2, we get:

24(-2) + 12 - 18(-2)

Now, let's simplify. 24 times -2 is -48. -18 times -2 is 36. So, our expression becomes:

-48 + 12 + 36

-48 + 12 is -36, and -36 + 36 is 0. So, the left side of the equation simplifies to 0.

Now, let's tackle the right side of the equation: 10 + 2p - 6. Again, we'll replace p with -2:

10 + 2(-2) - 6

2 times -2 is -4, so our expression becomes:

10 - 4 - 6

10 - 4 is 6, and 6 - 6 is 0. So, the right side of the equation also simplifies to 0.

We've shown that when we plug p = -2 into our original equation, both sides are equal to 0. This means our solution is correct! We've successfully verified that p = -2 is indeed the value that makes the equation true. High fives all around!

Conclusion

So, there you have it, folks! We've successfully solved for p in the linear equation 24p + 12 - 18p = 10 + 2p - 6. By following our step-by-step guide – simplifying both sides, isolating the variable term, solving for p, and verifying our solution – we've shown that p = -2. You guys are absolute algebra superstars!

Remember, solving linear equations is all about breaking down a problem into smaller, more manageable steps. Don't be afraid to take your time, show your work, and double-check your answers. With a little practice, you'll be solving equations like a pro in no time.

Keep up the awesome work, and we'll catch you in the next math adventure! Stay curious, stay creative, and never stop exploring the amazing world of mathematics. Peace out!