Solve For P: 82.78(p + 11.892) = 72.8464

Hey there, math whizzes and problem-solvers! Today, we're diving into a neat little algebraic puzzle that’s perfect for flexing those problem-solving muscles. We've got an equation, 82.78(p + 11.892) = 72.8464, and our mission, should we choose to accept it, is to find the exact value of p without any rounding. This isn't just about getting the right answer, guys; it’s about understanding the step-by-step process, the logic behind each move, and how to manipulate equations to isolate that elusive variable. So, grab your calculators, your notebooks, and maybe a comfy chair, because we're about to break this down together. We'll go through each step, explaining the 'why' behind the 'what', ensuring that by the end, you’ll not only have the solution but also a clearer grasp of algebraic manipulation. Get ready to conquer this equation and boost your confidence in tackling similar problems. Let's get started!

Understanding the Equation and Our Goal

Alright guys, before we jump headfirst into solving, let's take a moment to appreciate the equation we're dealing with: 82.78(p + 11.892) = 72.8464. Our primary objective here is to find the value of the variable p. Think of p as a mystery number that, when plugged into this equation, makes the left side exactly equal to the right side. The equation involves a number multiplied by a term in parentheses, and that term in parentheses contains our variable p along with a constant. Our strategy will involve a series of inverse operations to peel away the numbers surrounding p, isolating it on one side of the equals sign. The key here is that whatever we do to one side of the equation, we must do to the other to maintain the balance. We’re aiming for an exact answer, meaning no approximations or rounded figures – we want the precise mathematical value of p. This often involves working with fractions or decimals as they are, rather than converting them to less precise forms. So, our journey starts with recognizing the structure of the equation and clearly defining our target: solve for p. The coefficients and constants might look a bit intimidating with all those decimal places, but trust me, the process is straightforward if we take it one step at a time. We need to be careful with our arithmetic, especially when dealing with decimals, to ensure accuracy. This problem is a fantastic exercise in applying the order of operations in reverse and understanding the properties of equality. Let’s get ready to break it down, step by step, and reveal the value of p!

Step 1: Isolating the Parenthetical Term

Okay, team, the first move in our strategy to solve for p is to get the expression inside the parentheses, which is (p + 11.892), all by itself on one side of the equation. Right now, this entire term is being multiplied by 82.78. To undo multiplication, what do we use? That's right, division! So, we need to divide both sides of our equation by 82.78. Remember, whatever we do to one side, we must do to the other to keep things balanced. This is a fundamental rule in algebra, and it's crucial for getting the correct solution. Let's write it out:

82.78(p + 11.892) / 82.78 = 72.8464 / 82.78

On the left side, the 82.78 in the numerator and the denominator cancel each other out, leaving us with just (p + 11.892). Now, let's focus on the right side. We need to perform the division: 72.8464 divided by 82.78. This is where a calculator comes in handy, but it's important to input the numbers carefully. Let's crunch those numbers:

72.8464 / 82.78 = 0.88

So, after this first crucial step, our equation simplifies significantly. It now looks like this:

p + 11.892 = 0.88

See? We've already made great progress! We've successfully isolated the part of the equation that contains our variable p. This step is all about using inverse operations – in this case, division to undo multiplication – to simplify the equation and bring us closer to finding the value of p. It's like peeling back the layers of an onion, removing one layer of complexity at a time. Keep that calculator handy for the next step, and let's keep the momentum going!

Step 2: Isolating the Variable 'p'

Alright, mathletes, we’re in the home stretch to solve for p! We've successfully simplified our equation to p + 11.892 = 0.88. Now, our variable p is almost free – it just has 11.892 added to it. To get p completely by itself, we need to undo this addition. What's the inverse operation of addition? You guessed it – subtraction! We're going to subtract 11.892 from both sides of the equation. This maintains the equality, ensuring our solution remains accurate.

Let's apply this to our equation:

p + 11.892 - 11.892 = 0.88 - 11.892

On the left side, adding 11.892 and then subtracting 11.892 results in zero, leaving us with just p. Now, for the right side, we need to perform the subtraction: 0.88 minus 11.892.

Let's calculate that:

0.88 - 11.892 = -11.012

So, the result of this step gives us our final answer for p:

p = -11.012

And there you have it, folks! We've successfully isolated p and found its exact value without any rounding. This second step involved using the inverse operation of subtraction to undo the addition, completely isolating our variable. It’s a satisfying feeling when you can see the solution clearly emerge from the steps you’ve taken. Remember this process: identify what’s with the variable, use inverse operations to remove it, and always maintain the balance of the equation. High five for solving this one!

Verification: Plugging the Solution Back In

Alright, brilliant mathematicians, we’ve reached the finish line in our quest to solve for p, and we found that p = -11.012. But you know what makes a solution even better? Verification! It’s always a good idea, especially in math, to double-check our work. This means plugging our found value of p back into the original equation and seeing if both sides are indeed equal. This step is super important because it confirms that our calculations were accurate and that we truly found the correct value for p. It’s like a final seal of approval on our problem-solving efforts.

Our original equation was: 82.78(p + 11.892) = 72.8464

And we found: p = -11.012

Let’s substitute -11.012 for p in the equation:

82.78(-11.012 + 11.892) = 72.8464

First, let's solve the expression inside the parentheses: -11.012 + 11.892.

-11.012 + 11.892 = 0.88

Now, substitute this result back into our equation:

82.78(0.88) = 72.8464

Finally, let's perform the multiplication on the left side: 82.78 times 0.88.

82.78 * 0.88 = 72.8464

Look at that! 72.8464 = 72.8464.

The left side perfectly matches the right side! This confirms that our solution p = -11.012 is absolutely correct. This verification step is a powerful tool. It not only boosts your confidence in your answers but also helps you catch any potential arithmetic errors or logical missteps. So, next time you solve an equation, don't forget to plug your answer back in. It's a small step that makes a big difference in ensuring accuracy. Awesome job, everyone!

Conclusion: Mastering Algebraic Equations

And just like that, guys, we've successfully navigated the complexities of the equation 82.78(p + 11.892) = 72.8464 and found the exact solution for p, which is p = -11.012. We accomplished this by systematically applying the principles of algebra: using inverse operations to isolate the variable p. First, we divided both sides by 82.78 to remove the coefficient outside the parentheses, simplifying the equation to p + 11.892 = 0.88. Then, we subtracted 11.892 from both sides to finally isolate p, revealing its value as -11.012. The crucial verification step further solidified our confidence by showing that plugging this value back into the original equation resulted in a true statement (72.8464 = 72.8464).

This process isn't just about solving one specific problem; it's about building a foundational understanding of how algebraic equations work. Every time you tackle an equation like this, you're strengthening your ability to think logically, break down complex problems into manageable steps, and develop critical analytical skills. These skills are transferable to countless areas, far beyond mathematics. So, whether you're dealing with homework assignments, real-world problems, or just enjoy the mental challenge, remember the power of inverse operations, maintaining balance, and verifying your results. Keep practicing, keep exploring, and never shy away from a challenging equation. You've got this! Happy solving!