Solving -3(8k + 5) = 3(9 - K): A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little algebraic equation: -3(8k + 5) = 3(9 - k). Don't worry, it might look a bit intimidating at first, but we'll break it down step-by-step so it's super easy to understand. We'll use the core principles of algebraic manipulation to isolate our variable 'k' and find its value. Think of it like a puzzle – each step is a piece that fits perfectly to reveal the final solution. So, grab your thinking caps, and let's get started!

1. The Distributive Property: Unpacking the Equation

In solving equations, the distributive property is your best friend when you encounter parentheses. It's like unpacking a gift – you need to see what's inside! In our equation, -3(8k + 5) = 3(9 - k), we have two sets of parentheses, each requiring our attention. The distributive property states that a(b + c) = ab + ac. Essentially, you multiply the term outside the parentheses by each term inside. Let’s apply this to our equation.

First, let’s tackle the left side of the equation, -3(8k + 5). We multiply -3 by both 8k and 5. So, -3 * 8k gives us -24k, and -3 * 5 results in -15. Therefore, the left side becomes -24k - 15. Remember, guys, pay close attention to the signs! A negative multiplied by a positive gives a negative.

Now, let’s move to the right side of the equation, 3(9 - k). We multiply 3 by both 9 and -k. So, 3 * 9 is 27, and 3 * -k is -3k. Thus, the right side transforms into 27 - 3k. Again, watch those signs! A positive multiplied by a negative yields a negative. By applying the distributive property, we've successfully expanded our equation, making it easier to work with. Our equation now looks like this: -24k - 15 = 27 - 3k. See? Already less intimidating!

Why is this step so crucial? Well, without distributing, we can't combine like terms or isolate 'k'. The distributive property is the key to unlocking the rest of the solution. It's like the first domino in a chain reaction – once it falls, the rest follow smoothly. So, mastering this step is fundamental for solving more complex algebraic problems down the road. Keep practicing, and you'll become a pro in no time!

2. Combining Like Terms: Gathering the 'k's and Numbers

After distributing, the next strategic move in solving our equation, -24k - 15 = 27 - 3k, is to combine like terms. Think of it as sorting your socks – you want all the pairs together! In algebraic terms, 'like terms' are those that have the same variable raised to the same power (in this case, just 'k') or are constants (plain numbers without any variables).

Our goal is to gather all the 'k' terms on one side of the equation and all the constant terms on the other. This makes it easier to isolate 'k' later on. Let's start by moving the -3k term from the right side to the left side. To do this, we perform the inverse operation – we add 3k to both sides of the equation. Remember, whatever you do to one side, you must do to the other to maintain the balance! This gives us: -24k - 15 + 3k = 27 - 3k + 3k. Simplifying this, we get -21k - 15 = 27.

Now, let's gather the constant terms. We have -15 on the left side, which we want to move to the right side. To do this, we add 15 to both sides of the equation: -21k - 15 + 15 = 27 + 15. This simplifies to -21k = 42. We've successfully combined like terms, making our equation much cleaner and closer to the solution. Combining like terms is a fundamental step that streamlines the solving process. It’s like organizing your workspace before tackling a big project – it makes everything more manageable and efficient.

3. Isolating the Variable: Freeing 'k'

We're in the home stretch now! Our equation is currently at -21k = 42. The ultimate goal in solving any equation is to isolate the variable. This means getting 'k' all by itself on one side of the equation. Right now, 'k' is being multiplied by -21. So, to undo this multiplication, we need to perform the inverse operation: division.

We'll divide both sides of the equation by -21. Remember, maintaining balance is key! This gives us (-21k) / -21 = 42 / -21. On the left side, -21k divided by -21 just leaves us with 'k'. On the right side, 42 divided by -21 gives us -2. Therefore, we have k = -2. We've done it! We've successfully isolated 'k' and found its value.

Why is isolating the variable the final step? Because it gives us the solution! It tells us the value of 'k' that makes the original equation true. It's like finding the missing piece of a puzzle – once you have it, the picture is complete. Isolating the variable might seem like a small step, but it’s the culmination of all the previous steps. Without it, we wouldn't have our answer. So, remember to always keep your eye on the prize – isolating that variable!

4. Checking Your Solution: The Ultimate Sanity Check

Okay, we've solved for 'k' and found that k = -2. But before we do a victory dance, it's crucial to check our solution. This step is like proofreading a paper – it ensures we haven't made any silly mistakes along the way. Plugging our solution back into the original equation is the ultimate sanity check.

Our original equation was -3(8k + 5) = 3(9 - k). Let's substitute k = -2 into the equation: -3(8*(-2) + 5) = 3(9 - (-2)). Now, we need to simplify both sides and see if they are equal. First, let’s simplify the left side: -3(-16 + 5) = -3(-11) = 33. Now, let’s simplify the right side: 3(9 + 2) = 3(11) = 33. Guess what? Both sides equal 33! This confirms that our solution, k = -2, is correct.

Why is checking your solution so important? Well, imagine building a house and skipping the inspection – you might end up with a shaky foundation! Checking your solution in math is the same idea. It verifies that your answer is accurate and that you haven’t made any errors in your calculations. It’s especially useful in exams or when working on complex problems. It gives you confidence in your answer and helps you avoid losing points for simple mistakes. So, make checking your solution a habit, guys! It's the secret weapon of every successful equation solver.

Conclusion: You Did It!

Fantastic job, guys! We've successfully solved the equation -3(8k + 5) = 3(9 - k), step by step. We used the distributive property, combined like terms, isolated the variable, and even checked our solution to make sure we nailed it. Remember, math might seem daunting at times, but breaking it down into manageable steps makes it much easier. So, keep practicing, keep exploring, and you'll become an equation-solving superstar in no time! You've got this!