Solving The Equation: 5x - (x + 3) = 3(2 - 2x) + X
Hey guys! Today, we're diving into a fun math problem: solving the equation 5x - (x + 3) = 3(2 - 2x) + x. Don't worry, it might look a little intimidating at first, but we're going to break it down step by step so it's super easy to understand. Whether you're a student tackling algebra homework or just someone who enjoys a good math challenge, this guide is for you. We'll go through each step meticulously, ensuring you grasp the logic behind every move. So, grab your pencil and paper, and let's get started on this mathematical adventure!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation is telling us. We've got variables (that's the 'x'), numbers, and some operations all mixed together. The goal is to find the value of 'x' that makes the equation true. In simpler terms, we want to figure out what number 'x' needs to be so that both sides of the equals sign are the same. To kick things off, we need to simplify both sides of the equation. This involves getting rid of those parentheses and combining like terms. Think of it as decluttering before we start organizing. By simplifying, we'll make the equation much easier to work with and reduce the chances of making mistakes along the way. So, let's roll up our sleeves and dive into the first steps of simplification!
Breaking Down the Left-Hand Side
Okay, let's start with the left side of the equation: 5x - (x + 3). The first thing we need to tackle is those parentheses. Remember, when we have a minus sign in front of parentheses, it's like we're multiplying everything inside by -1. So, we're going to distribute that negative sign to both the 'x' and the '3' inside the parentheses. This means the 'x' becomes '-x' and the '+3' becomes '-3'. Our expression now looks like this: 5x - x - 3. See how we've essentially flipped the signs of the terms inside the parentheses? Now, we can combine the 'x' terms. We have '5x' and '-x', which together give us '4x'. So, the simplified left-hand side is 4x - 3. We've successfully decluttered this side of the equation, making it much simpler to handle. This is a crucial step because it sets us up for easier manipulation of the equation later on. Let's move on to the right-hand side and see what simplifications we can make there!
Simplifying the Right-Hand Side
Now, let's tackle the right-hand side of the equation: 3(2 - 2x) + x. Here, we have a number multiplying a set of parentheses, and then an additional 'x' term hanging out at the end. Just like before, we need to distribute to get rid of those parentheses. This time, we're distributing the '3' across both the '2' and the '-2x' inside the parentheses. So, we multiply 3 by 2, which gives us 6, and we multiply 3 by -2x, which gives us -6x. Our expression now looks like this: 6 - 6x + x. Great! We've cleared the parentheses. Now, just like on the left-hand side, we can combine like terms. We have '-6x' and '+x', which combine to give us '-5x'. So, the simplified right-hand side is 6 - 5x. We've successfully transformed the right side into a more manageable form. With both sides of the equation simplified, we're in a much better position to solve for 'x'. Let's move on to the next step and bring those 'x' terms together!
Isolating the Variable
Alright, now that we've simplified both sides, our equation looks like this: 4x - 3 = 6 - 5x. The next big step is to get all the 'x' terms on one side of the equation and all the constant terms (the numbers without 'x') on the other side. This is what we call isolating the variable. Think of it like sorting your laundry – you want to group all the socks together and all the shirts together. To do this, we'll use inverse operations. This means we'll add or subtract terms on both sides of the equation to move them around. The key is to do the same thing to both sides to keep the equation balanced. If we add 5 to one side, we must add 5 to the other. If we subtract 2x from one side, we must subtract 2x from the other. It's all about maintaining that equality! So, let's start moving those terms around and get closer to solving for 'x'.
Moving the 'x' Terms
Let's start by moving all the 'x' terms to the left side of the equation. We have 4x - 3 = 6 - 5x. Currently, we have a '-5x' term on the right side, and we want to get rid of it from there and bring it over to the left. To do this, we'll use the inverse operation: addition. We're going to add '5x' to both sides of the equation. This is because adding '5x' to '-5x' will cancel it out on the right side, leaving us with just the constant term. So, we add '5x' to both sides:
(4x - 3) + 5x = (6 - 5x) + 5x
On the left side, '4x' plus '5x' gives us '9x', so the left side becomes 9x - 3. On the right side, '-5x' plus '5x' cancels out, leaving us with just '6'. So, the equation now looks like this:
9x - 3 = 6
Awesome! We've successfully moved all the 'x' terms to the left side. Now, let's tackle the constant terms and get them all on the right side.
Shifting the Constants
Now that we have 9x - 3 = 6, let's shift the constant terms. We want to get the '-3' on the left side over to the right side with the other constant, '6'. Again, we'll use the inverse operation. Since we have '-3' on the left, we'll add '3' to both sides of the equation. This will cancel out the '-3' on the left and move it over to the right side:
(9x - 3) + 3 = 6 + 3
On the left side, '-3' plus '3' cancels out, leaving us with just 9x. On the right side, '6' plus '3' gives us '9'. So, the equation now looks like this:
9x = 9
Fantastic! We've isolated the 'x' term on one side and the constant term on the other. We're in the home stretch now. The last step is to solve for 'x' by getting it all by itself.
Solving for 'x'
We've reached the final stage! Our equation is now 9x = 9. This means we have 9 times 'x' equals 9. To find the value of 'x', we need to undo that multiplication. And how do we undo multiplication? That's right, we use division! We're going to divide both sides of the equation by the number that's multiplying 'x', which in this case is '9'. This will isolate 'x' and give us our answer. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. So, let's divide both sides by 9 and see what we get.
Dividing Both Sides
Okay, let's divide both sides of the equation 9x = 9 by 9. This looks like:
(9x) / 9 = 9 / 9
On the left side, we have '9x' divided by '9'. The '9's cancel each other out, leaving us with just 'x'. On the right side, we have '9' divided by '9', which equals 1. So, the equation simplifies to:
x = 1
And there you have it! We've solved for 'x'. The value of 'x' that makes the original equation true is 1. It's like we've cracked the code and found the secret number that balances the equation. Now, to be absolutely sure we've got it right, it's always a good idea to check our answer.
Checking the Solution
We've found that x = 1 is the solution to our equation. But before we declare victory, let's double-check our work to make sure we haven't made any sneaky mistakes along the way. The best way to do this is to plug our solution back into the original equation and see if it holds true. If both sides of the equation are equal when we substitute 'x = 1', then we know we've got the right answer. It's like putting the key in the lock and making sure it turns smoothly. So, let's take our solution and substitute it back into the original equation: 5x - (x + 3) = 3(2 - 2x) + x
Plugging in x = 1
Let's substitute x = 1 into the original equation: 5x - (x + 3) = 3(2 - 2x) + x. Replacing each 'x' with '1', we get:
5(1) - (1 + 3) = 3(2 - 2(1)) + 1
Now, we need to simplify both sides of the equation following the order of operations (PEMDAS/BODMAS). Let's start with the left side. We have:
5(1) - (1 + 3)
First, we solve the parentheses: 1 + 3 = 4. So, the left side becomes:
5(1) - 4
Next, we do the multiplication: 5(1) = 5. So, the left side simplifies to:
5 - 4
Finally, we do the subtraction: 5 - 4 = 1. So, the left side of the equation equals 1.
Now, let's simplify the right side of the equation:
3(2 - 2(1)) + 1
First, we solve the innermost parentheses: 2(1) = 2. So, the right side becomes:
3(2 - 2) + 1
Next, we solve the remaining parentheses: 2 - 2 = 0. So, the right side becomes:
3(0) + 1
Now, we do the multiplication: 3(0) = 0. So, the right side simplifies to:
0 + 1
Finally, we do the addition: 0 + 1 = 1. So, the right side of the equation equals 1.
Both sides of the equation equal 1 when we substitute x = 1. This means our solution is correct! We've successfully solved and verified the equation. Give yourself a pat on the back!
Conclusion
Fantastic job, guys! We've successfully navigated the equation 5x - (x + 3) = 3(2 - 2x) + x, step by step, and found that the solution is x = 1. We started by understanding the equation, then simplified both sides by distributing and combining like terms. Next, we isolated the variable by moving all the 'x' terms to one side and the constants to the other. Finally, we solved for 'x' by dividing both sides by the coefficient of 'x'. And, just to be sure, we checked our solution by plugging it back into the original equation and verifying that both sides were equal. Solving equations like this is a fundamental skill in algebra, and you've now added another tool to your mathematical toolkit. Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to explore more equations, feel free to ask. Happy solving!
