Solving For X: -9x - 44 = -7(x + 4) - Step-by-Step Guide
Hey guys! Today, we're diving into a common algebra problem: solving for x in the equation -9x - 44 = -7(x + 4). Don't worry, it might look intimidating at first, but we'll break it down step by step so you can conquer these types of problems with confidence. So, grab your pencils and notebooks, and let’s get started!
Understanding the Equation
Before we jump into solving, let’s make sure we understand what the equation is telling us. In the equation -9x - 44 = -7(x + 4), our mission is to isolate 'x' on one side of the equation. This means we want to manipulate the equation using mathematical operations until we have 'x' by itself on either the left or right side. The key principle here is that whatever we do to one side of the equation, we must do to the other side to maintain balance. Think of it like a scale – if you add weight to one side, you need to add the same weight to the other side to keep it level.
When we say “solve for x,” what we're really trying to find is the value of 'x' that makes the equation true. In other words, we want to find the number that, when substituted for 'x', will make both sides of the equation equal. This is a fundamental concept in algebra, and mastering it will open the door to solving more complex problems. Remember, guys, algebra is all about finding the unknowns, and 'x' is often our main character in this quest. So let’s get ready to put on our detective hats and find the value of this mysterious 'x'!
The Importance of Order of Operations
One crucial thing to remember when solving equations like this is the order of operations. You might have heard of it as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Both acronyms serve as a guide to help us tackle mathematical problems in the correct sequence. Imagine trying to build a house without following the blueprint – you might end up with something a bit wonky! Similarly, skipping steps or performing operations out of order can lead to the wrong answer.
In our equation, -9x - 44 = -7(x + 4), we'll need to deal with the parentheses first, then any multiplication or division, and finally addition and subtraction. This ensures that we're unraveling the equation in a systematic way, like carefully unwrapping a gift layer by layer. Mastering the order of operations is like having a secret weapon in your mathematical toolkit – it will help you navigate complex equations with precision and confidence. So let's keep PEMDAS or BODMAS in mind as we move forward, guys, and we'll be solving equations like pros in no time!
Step-by-Step Solution
Alright, guys, let's dive into the step-by-step solution for the equation -9x - 44 = -7(x + 4). We'll break it down into manageable chunks, making sure each step is crystal clear. Remember, the goal here is to isolate 'x' on one side of the equation.
Step 1: Distribute
The first thing we need to do is tackle those parentheses. We have -7(x + 4) on the right side of the equation. To get rid of the parentheses, we'll use the distributive property. This means we multiply the -7 by each term inside the parentheses. So, -7 * x gives us -7x, and -7 * 4 gives us -28. Now, our equation looks like this: -9x - 44 = -7x - 28.
Distributing is a fundamental skill in algebra, and it's like unlocking a door to simplify the equation. By multiplying the term outside the parentheses by each term inside, we're essentially spreading the operation evenly. This step is crucial because it allows us to combine like terms later on. Imagine if we didn't distribute – we'd be stuck with those parentheses, and it would be much harder to isolate 'x'. So, remember guys, whenever you see parentheses in an equation, distribution is often the first key step to take!
Step 2: Move the x Terms
Now that we've distributed, let's gather all the 'x' terms on one side of the equation. We have -9x on the left and -7x on the right. A good strategy here is to move the smaller 'x' term to the side with the larger 'x' term to avoid dealing with negative coefficients. In this case, -9x is smaller than -7x, so we'll move the -7x term from the right side to the left side. To do this, we add 7x to both sides of the equation:
-9x - 44 + 7x = -7x - 28 + 7x
This simplifies to:
-2x - 44 = -28
Moving the 'x' terms is like herding sheep – we want to get all the 'x's together in one place! By adding 7x to both sides, we're effectively canceling out the -7x on the right and bringing it over to the left side. This is a crucial step because it brings us closer to isolating 'x'. Remember guys, the goal is to get 'x' all by itself, so we need to gather all the terms containing 'x' on one side. By strategically moving the 'x' terms, we're making our job of solving for 'x' much easier!
Step 3: Move the Constants
We're making great progress, guys! Now that we have all the 'x' terms on one side, let's move the constants (the numbers without 'x') to the other side. We have -2x - 44 = -28. We want to isolate the -2x term, so we need to get rid of the -44 on the left side. To do this, we'll add 44 to both sides of the equation:
-2x - 44 + 44 = -28 + 44
This simplifies to:
-2x = 16
Moving the constants is like clearing the stage for our star, 'x'! By adding 44 to both sides, we're canceling out the -44 on the left and bringing it over to the right side as a positive 44. This leaves us with -2x isolated on the left, which is exactly what we want. Remember, guys, constants are like the supporting cast in our equation drama – they play an important role, but ultimately, we want to focus on our main character, 'x'. By strategically moving the constants, we're setting the stage for the final act of solving for 'x'!
Step 4: Isolate x
We're in the home stretch now, guys! We have -2x = 16. To finally isolate 'x', we need to get rid of the -2 that's multiplying it. To do this, we'll divide both sides of the equation by -2:
-2x / -2 = 16 / -2
This simplifies to:
x = -8
And there you have it! We've solved for x. Dividing both sides by -2 is like performing the final magic trick – it reveals the value of 'x'. By dividing, we're undoing the multiplication that was happening between -2 and x. This leaves us with x all by itself, proudly displaying its value. Remember guys, the ultimate goal of solving for x is to isolate it, and dividing by the coefficient is often the final step in this process. So, with a bit of algebraic wizardry, we've successfully found that x = -8!
Checking Your Answer
Alright, guys, we've solved for x, but how do we know if our answer is correct? It's always a good idea to check your solution, especially in algebra. Think of it as proofreading your work – you want to make sure you haven't made any mistakes along the way.
The Substitution Method
The best way to check our answer is to substitute the value we found for 'x' back into the original equation. In our case, we found that x = -8. So, we'll plug -8 in for 'x' in the original equation: -9x - 44 = -7(x + 4).
Substituting x = -8, we get:
-9(-8) - 44 = -7(-8 + 4)
Now, we simplify both sides of the equation following the order of operations (PEMDAS/BODMAS).
Simplify Both Sides
Let's start with the left side:
-9(-8) - 44 = 72 - 44 = 28
Now, let's simplify the right side:
-7(-8 + 4) = -7(-4) = 28
Verify the Equality
We've simplified both sides, and guess what? Both sides equal 28! This means our solution, x = -8, is correct. Hooray! Checking our answer is like putting the puzzle pieces together to see if they fit – when the equation balances out, we know we've found the right value for 'x'. Remember guys, always double-check your work, and substitution is your best friend in the world of algebra!
Common Mistakes to Avoid
Alright, guys, now that we've successfully solved for 'x' and checked our answer, let's talk about some common pitfalls that students often encounter when tackling these types of equations. Knowing these mistakes can help you steer clear of them and boost your confidence in solving algebraic problems.
Distribution Errors
One of the most frequent errors occurs during the distribution step. Remember, when you have a term outside parentheses, you need to multiply it by every term inside the parentheses. For example, in our equation -9x - 44 = -7(x + 4), it's crucial to multiply -7 by both x and 4. A common mistake is to only multiply by the first term, forgetting the second. So, make sure you're distributing carefully and thoroughly, guys!
Sign Errors
Another common mistake involves sign errors. When dealing with negative numbers, it's easy to slip up. For instance, when moving terms across the equals sign, remember to change the sign. If a term is being subtracted on one side, it becomes addition on the other side, and vice versa. Also, watch out for multiplying or dividing by negative numbers – a negative times a negative is a positive, and a positive times a negative is a negative. Keep those sign rules in mind, guys, and you'll avoid a lot of unnecessary headaches!
Order of Operations
We've already emphasized the importance of the order of operations (PEMDAS/BODMAS), but it's worth mentioning again. Skipping steps or performing operations out of order can lead to incorrect results. Make sure you're addressing parentheses first, then exponents, multiplication and division, and finally addition and subtraction. Sticking to the correct order is like following a recipe – it ensures a delicious outcome (in this case, a correct solution!). Remember guys, PEMDAS/BODMAS is your trusty guide in the world of math!
Practice Problems
Alright, guys, now that we've walked through the solution, checked our answer, and discussed common mistakes, it's time to put your skills to the test! Practice is key to mastering algebra, so let's tackle a few more problems similar to the one we just solved. Grab your notebooks and pencils, and let's get to work!
Problem 1: 5(x - 3) = 2x + 6
Problem 2: -3(2x + 1) = -15
Problem 3: 4x - 7 = -2(x + 5)
Remember guys, the more you practice, the more comfortable and confident you'll become with solving these types of equations. Don't be afraid to make mistakes – they're a natural part of the learning process. Just take your time, break down the problems step by step, and remember the strategies we've discussed. You've got this!
Conclusion
So there you have it, guys! We've successfully navigated the equation -9x - 44 = -7(x + 4), breaking it down into manageable steps and conquering it with confidence. We started by understanding the equation and the importance of isolating 'x'. Then, we walked through the step-by-step solution, distributing, moving 'x' terms and constants, and finally isolating 'x'. We didn't stop there – we also checked our answer using substitution, ensuring our solution was correct. And to top it off, we discussed common mistakes to avoid and tackled some practice problems.
Remember, guys, solving algebraic equations is like building a house – each step is a crucial part of the foundation. By mastering these fundamental skills, you're setting yourself up for success in more advanced math topics. So keep practicing, stay persistent, and don't be afraid to ask for help when you need it. You've got the tools and the knowledge – now go out there and conquer those equations! Keep up the great work!
