Solve 5 X ( P + 9 ) - 13 = 57: A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of numbers and letters? Don't worry, we've all been there! Today, we're going to break down the equation 5 x ( P + 9 ) - 13 = 57 step by step, making it super easy to understand and solve. Think of it like a puzzle – each step we take gets us closer to finding the missing piece, which in this case, is the value of 'P'. So, grab your thinking caps, and let's dive in!

Cracking the Code: A Step-by-Step Solution

Our main goal here is to isolate 'P' on one side of the equation. This means we need to undo all the operations that are happening to 'P'. Remember the order of operations? It's like a secret code: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). We'll be working backward through this order to solve for 'P'.

1. Undoing Subtraction: The first thing we see on the left side is '- 13'. To get rid of it, we need to do the opposite operation, which is addition. So, we add 13 to both sides of the equation. This is crucial – whatever we do to one side, 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 weight to the other to keep it level.

   Our equation now looks like this:

   5 x ( P + 9 ) - 13 + 13 = 57 + 13

   Simplifying, we get:

   5 x ( P + 9 ) = 70

2. Undoing Multiplication: Next up, we have '5' multiplied by the parentheses. To undo this, we divide both sides of the equation by 5. Remember, division is the opposite of multiplication.

   This gives us:

   (5 x ( P + 9 )) / 5 = 70 / 5

   Simplifying, we have:

   P + 9 = 14

3. Undoing Addition: We're almost there! Now we have 'P + 9'. To isolate 'P', we need to undo the addition by subtracting 9 from both sides. Subtraction is the opposite of addition, just like we used addition to undo subtraction earlier.

   So, we get:

   P + 9 - 9 = 14 - 9

   Simplifying, we finally find:

   P = 5

Boom! We've cracked the code! The value of 'P' that makes the equation true is 5. But hold on, we're not done yet. It's always a good idea to double-check our answer to make sure we didn't make any sneaky mistakes along the way. Think of it as proofreading your work before you hand it in.

Double-Checking Our Solution: The Verification Process

To verify our solution, we'll substitute 'P = 5' back into the original equation and see if both sides are equal. This is like plugging the missing puzzle piece into the puzzle – if it fits perfectly, we know we've got the right answer.

Our original equation was:

5 x ( P + 9 ) - 13 = 57

Now, let's replace 'P' with 5:

5 x ( 5 + 9 ) - 13 = 57

Now we follow the order of operations (PEMDAS/BODMAS) to simplify the left side:

1. Parentheses: 5 + 9 = 14

   So, we have:

   5 x 14 - 13 = 57

2. Multiplication: 5 x 14 = 70

   Now we have:

   70 - 13 = 57

3. Subtraction: 70 - 13 = 57

   Finally, we get:

   57 = 57

Ta-da! The left side of the equation equals the right side. This confirms that our solution, P = 5, is indeed correct. We've not only solved the equation but also verified our answer, making sure we're 100% confident in our result. That's the power of checking your work, guys! It's like having a safety net that catches any potential errors before they become a problem.

Understanding the Underlying Principles: Why This Works

Now that we've solved the equation, let's take a step back and think about why these steps work. It's not just about following a set of rules; it's about understanding the fundamental principles of algebra. The key concept here is the idea of balance. An equation is like a perfectly balanced scale. The left side must always equal the right side. Any operation we perform on one side, we must perform on the other to maintain this balance.

Think of it like this: if you add a weight to one side of the scale, you need to add the same weight to the other side to keep it level. Similarly, in an equation, if you add, subtract, multiply, or divide on one side, you need to do the exact same thing on the other side. This ensures that the equation remains true and that the two sides remain equal.

This principle of balance is the foundation of algebra. It allows us to manipulate equations and isolate the variable we're trying to solve for. Each step we took in solving the equation – adding 13, dividing by 5, subtracting 9 – was designed to undo the operations that were happening to 'P' while maintaining the balance of the equation. By carefully applying these principles, we can confidently solve even the most complex-looking equations.

Furthermore, understanding the order of operations (PEMDAS/BODMAS) is crucial. It's like having a roadmap for solving equations. It tells us the correct sequence to perform operations so that we arrive at the correct answer. We work backward through the order of operations when solving for a variable, undoing each operation in reverse order.

Real-World Applications: Where Equations Come to Life

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