Solving For P: A Step-by-Step Guide To -160 = 22 + 8(3p - 4)

Hey guys! Today, we're diving into a math problem where we need to solve for p in the equation -160 = 22 + 8(3p - 4). Don't worry; it might look intimidating at first, but we'll break it down step by step so it's super easy to follow. Think of this as a puzzle, and we're the detectives figuring out the value of p. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into solving, let's understand what this equation is all about. We have an algebraic equation, which means we have variables (p in this case) and constants (-160, 22, 8, 3, and -4) connected by mathematical operations. Our goal is to isolate p on one side of the equation to find out what value it holds. This involves using the order of operations in reverse (PEMDAS/BODMAS) and applying inverse operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to help solve it accurately. Let's break down each part of the equation:

• -160: This is a constant term on the left side of the equation. It's a fixed number and doesn't change.
• 22: Another constant term, this time on the right side of the equation. Like -160, it's a fixed value.
• 8(3p - 4): This part is where the variable p comes in. We have a term in parentheses (3p - 4) being multiplied by 8. This is where we'll need to use the distributive property later on. The key here is to recognize that this entire expression needs to be simplified before we can further isolate p. The term inside the parentheses involves both multiplication (3 times p) and subtraction (subtracting 4). Understanding this structure helps us plan our attack to solve for p.

By understanding each component, we set ourselves up for success in solving the equation. The equation is essentially a balance, and whatever we do on one side, we must do on the other to maintain that balance. So, with this understanding, let's move on to the first steps in solving for p!

Step 1: Distribute the 8

Okay, the first thing we need to do is tackle that 8(3p - 4) part. This is where the distributive property comes into play. Remember, the distributive property tells us that we need to multiply the 8 by both terms inside the parentheses. So, we're going to multiply 8 by 3p and then multiply 8 by -4. It's like giving 8 a little knock on the door of both terms inside the parentheses. Think of it this way: 8 is saying, "I want to multiply myself by everyone inside!"

Here's how it looks:

8 * (3p) = 24p 8 * (-4) = -32

So, 8(3p - 4) becomes 24p - 32. Now, let’s rewrite the entire equation with this simplification:

-160 = 22 + 24p - 32

See? We've gotten rid of the parentheses and made the equation a bit simpler. The distributive property is super important in algebra, guys, so make sure you're comfortable with it. It’s like having a superpower that lets you break down complex expressions into more manageable parts. By distributing the 8, we've cleared a major hurdle and brought ourselves closer to isolating p. Next up, we’ll combine the constant terms on the right side to further simplify things.

Step 2: Combine Like Terms

Alright, now that we've distributed the 8, our equation looks like this: -160 = 22 + 24p - 32. The next step is to combine the like terms on the right side of the equation. Like terms are just terms that have the same variable raised to the same power (or, in this case, just constant terms). So, we're looking at the 22 and the -32. Think of it like grouping all the apples together and all the oranges together – we’re just putting similar numbers together to make things easier.

We have 22 and -32. To combine them, we simply add them together:

22 + (-32) = -10

So, our equation now becomes:

-160 = -10 + 24p

We've reduced the number of terms on the right side, making the equation even simpler. Combining like terms is a crucial step in solving equations because it helps to tidy things up and makes the next steps more straightforward. It’s like organizing your workspace before tackling a big project – a little bit of organization can make a huge difference! We're getting closer and closer to isolating p, so keep up the great work! Next, we'll move those constant terms away from the p term.

Step 3: Isolate the Variable Term

Okay, we're making fantastic progress! Our equation is currently -160 = -10 + 24p. Now, we need to isolate the variable term, which is the 24p part. This means we want to get the 24p all by itself on one side of the equation. To do this, we need to get rid of the -10 that's hanging out with it on the right side. Remember, we need to perform the inverse operation to move terms around.

Since -10 is being added to 24p, we need to subtract -10 from both sides or, more simply, add 10 to both sides. Adding 10 will cancel out the -10 on the right side, leaving us with just the variable term. Remember, whatever we do to one side of the equation, we must do to the other to keep the equation balanced. It's like a seesaw – if you add weight to one side, you need to add the same amount to the other side to keep it level.

Here’s how it looks:

-160 + 10 = -10 + 24p + 10

Simplifying both sides, we get:

-150 = 24p

Awesome! We've successfully isolated the variable term. The equation is looking much simpler now, and we’re just one step away from finding the value of p. Next, we’ll deal with the coefficient of p to finally solve for our variable. Hang in there; you’re doing great!

Step 4: Solve for p

Alright, we're in the home stretch! Our equation now reads -150 = 24p. We’re so close to finding out what p equals! Remember, our goal is to get p all by itself on one side of the equation. Currently, p is being multiplied by 24. To undo this multiplication, we need to perform the inverse operation, which is division. Think of it as untangling a knot – we need to do the opposite of what's tying p up.

So, we're going to divide both sides of the equation by 24. This will cancel out the 24 on the right side, leaving us with p alone. Remember, we have to do the same thing to both sides to keep the equation balanced. It's like sharing a pizza – if you cut one slice in half, you need to cut all the slices in half to keep everything fair.

Here’s the math:

-150 / 24 = (24p) / 24

Simplifying, we get:

p = -150 / 24

Now, we can simplify the fraction -150/24. Both numbers are divisible by 6, so let's divide both the numerator and the denominator by 6:

-150 / 6 = -25 24 / 6 = 4

So, p = -25/4

We can also express this as a mixed number or a decimal. As a mixed number, -25/4 is -6 1/4. As a decimal, it’s -6.25.

Congratulations! We’ve solved for p. It's like cracking a code – we used our math skills to unlock the value of p. You guys did an amazing job sticking with it, and now you know how to solve equations like this one. Next, we'll do a quick check to make sure our answer is correct.

Step 5: Check Your Solution

Awesome job solving for p! We found that p = -25/4 (or -6.25). But before we celebrate too hard, it’s super important to check our solution. This is like proofreading an essay or double-checking your work – it ensures we didn't make any little mistakes along the way. Checking our solution gives us confidence that we got the right answer. It’s like having a secret weapon against errors!

To check our solution, we're going to plug the value we found for p back into the original equation. If both sides of the equation are equal after we substitute p, then we know our solution is correct. If they're not equal, we need to go back and see where we might have made a mistake. It’s like fitting a key into a lock – if it fits, you've got the right key; if it doesn't, you need to try another one.

Our original equation was:

-160 = 22 + 8(3p - 4)

Now, let's substitute p = -25/4 into the equation:

-160 = 22 + 8(3*(-25/4) - 4)

First, let’s simplify inside the parentheses:

3 * (-25/4) = -75/4

Now, we have:

-160 = 22 + 8(-75/4 - 4)

Let's convert 4 to a fraction with a denominator of 4: 4 = 16/4

Now, we can combine the terms inside the parentheses:

-75/4 - 16/4 = -91/4

So, our equation becomes:

-160 = 22 + 8(-91/4)

Next, let’s multiply 8 by -91/4:

8 * (-91/4) = -728/4 = -182

Now, our equation is:

-160 = 22 - 182

Finally, let's combine the terms on the right side:

22 - 182 = -160

So, we have:

-160 = -160

Yay! Both sides of the equation are equal, which means our solution p = -25/4 is correct! It's like hearing that satisfying click when the key fits perfectly into the lock. You’ve not only solved the equation but also confirmed that your solution is accurate. Give yourself a pat on the back – you've earned it!

Conclusion

And there you have it! We've successfully solved for p in the equation -160 = 22 + 8(3p - 4). We walked through each step, from distributing the 8 to combining like terms, isolating the variable, and finally solving for p. And, most importantly, we checked our solution to make sure it was correct. Solving equations like this is a fundamental skill in algebra, and you’ve just added another tool to your math toolbox. You guys are math superstars!

Remember, the key to solving algebraic equations is to take it one step at a time, stay organized, and always double-check your work. Math might seem tricky sometimes, but with practice and a bit of patience, you can conquer any problem. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!