Solving The Equation: X + X − 4 = 17 ⋅ 2x + 1
Hey guys! Let's dive into solving this math equation together. We've got x + x − 4 = 17 ⋅ 2x + 1, and our mission is to find the value of 'x' that makes this equation true. This involves a bit of algebraic maneuvering, so let’s break it down step by step. We’ll cover everything from simplifying the equation to isolating 'x' and, of course, verifying our solution. So, grab your pencils, and let’s get started!
Understanding the Equation
Before we jump into solving, it’s super important to understand what the equation is telling us. We have variables (that's 'x'), constants (like 4 and 1), and operations (addition, subtraction, and multiplication). The equal sign (=) tells us that whatever is on the left side has the same value as what’s on the right side. Think of it like a balance scale – we need to keep both sides balanced to find the correct value for 'x'.
In our equation, x + x − 4 = 17 ⋅ 2x + 1, the left side has two 'x' terms, a subtraction, and a constant. The right side involves multiplication of 'x' by 2, multiplication by 17, and addition. Our job is to simplify each side and then use inverse operations to isolate 'x'. This might sound intimidating, but trust me, we'll take it one step at a time and it'll all make sense.
Recognizing the different components—variables, constants, and operations—is the first key step. Once we identify these, we can start thinking about how to rearrange and simplify the equation. For example, combining like terms is a fundamental move in solving equations, and we'll definitely be using that here. So, let’s move on to the first step in solving this equation: simplifying both sides.
Step 1: Simplifying Both Sides
The first thing we want to do is simplify both sides of the equation as much as possible. This makes the equation easier to work with and reduces the chance of making mistakes. On the left side, we have x + x − 4. Notice that we have two 'x' terms. We can combine these like terms by adding them together. So, x + x becomes 2x. Now the left side looks like 2x − 4.
On the right side, we have 17 ⋅ 2x + 1. Here, we need to perform the multiplication first, according to the order of operations (PEMDAS/BODMAS). So, 17 multiplied by 2x is 34x. Now the right side simplifies to 34x + 1. See how much cleaner things look already?
Now our equation looks like this: 2x − 4 = 34x + 1. We’ve taken the original equation and simplified both sides by combining like terms and performing multiplication. This is a crucial step because it sets us up for the next phase, which involves isolating the variable 'x'. By simplifying, we’ve reduced the number of terms we need to deal with, making the algebra a lot more manageable.
Simplifying both sides is all about making the equation as straightforward as possible. It's like tidying up before you start a big project – it helps you stay organized and focused. Next up, we'll start moving terms around to get all the 'x' terms on one side and the constants on the other. Let's keep the momentum going!
Step 2: Moving Terms Around
Okay, now that we've simplified both sides, it's time to get strategic about moving terms around. Our goal is to get all the 'x' terms on one side of the equation and all the constants (the numbers) on the other side. This process involves using inverse operations, which means doing the opposite of what's currently being done to a term.
Looking at our simplified equation, 2x − 4 = 34x + 1, we have 'x' terms on both sides (2x and 34x) and constants on both sides (-4 and 1). A good first move is to subtract 2x from both sides. Why? Because it eliminates the 'x' term on the left side. Remember, what we do to one side, we must do to the other to keep the equation balanced. So, let's subtract 2x from both sides:
2x − 4 − 2x = 34x + 1 − 2x
This simplifies to −4 = 32x + 1. See how the 2x on the left side canceled out? Great! Now we have 'x' only on the right side. The next step is to get rid of the constant (+1) on the right side. To do this, we subtract 1 from both sides:
−4 − 1 = 32x + 1 − 1
This gives us −5 = 32x. We’re getting closer! Now we have all the 'x' terms on one side and all the constants on the other. The only thing left to do is to isolate 'x' completely. So, let's move on to the final step: isolating 'x'.
Step 3: Isolating 'x'
We've reached the final stretch! We now have the equation −5 = 32x. Our mission is to get 'x' all by itself on one side of the equation. Right now, 'x' is being multiplied by 32. To undo this multiplication, we need to perform the inverse operation, which is division. So, we'll divide both sides of the equation by 32.
−5 / 32 = (32x) / 32
On the right side, 32x divided by 32 just leaves us with 'x'. On the left side, we have −5 divided by 32, which we can write as a fraction: −5/32. So, our equation now looks like this:
−5/32 = x
That’s it! We’ve isolated 'x'. Our solution is x = −5/32. This might look like a strange fraction, but it’s perfectly valid. It means that if we substitute −5/32 for 'x' in the original equation, both sides should be equal. But before we declare victory, it's always a good idea to double-check our work. Let's move on to the crucial step of verifying our solution.
Step 4: Verifying the Solution
Alright, we’ve found a solution: x = −5/32. But before we celebrate, we need to make sure our solution is correct. The best way to do this is to substitute our value for 'x' back into the original equation and see if both sides come out equal. This is like the ultimate test of our algebraic skills!
Our original equation was x + x − 4 = 17 ⋅ 2x + 1. Let's plug in −5/32 for every 'x' we see:
(−5/32) + (−5/32) − 4 = 17 ⋅ 2(−5/32) + 1
This looks a bit messy, but don’t worry, we'll take it step by step. First, let's simplify the left side. We have (−5/32) + (−5/32), which is −10/32. We can simplify this fraction to −5/16. Now the left side looks like this:
−5/16 − 4
To subtract 4, we need to express it as a fraction with a denominator of 16. So, 4 is the same as 64/16. Thus, we have:
−5/16 − 64/16 = −69/16
So, the left side simplifies to −69/16. Now, let’s tackle the right side. We have 17 ⋅ 2(−5/32) + 1. First, let's multiply 2 by −5/32, which gives us −10/32. We can simplify this to −5/16. Now the right side looks like this:
17 ⋅ (−5/16) + 1
Next, we multiply 17 by −5/16, which gives us −85/16. So, we have:
−85/16 + 1
To add 1, we need to express it as a fraction with a denominator of 16. So, 1 is the same as 16/16. Thus, we have:
−85/16 + 16/16 = −69/16
Guess what? The right side also simplifies to −69/16! This means our solution is correct. Both sides of the equation are equal when we substitute x = −5/32. Woo-hoo! We did it!
Conclusion
So, guys, we've successfully solved the equation x + x − 4 = 17 ⋅ 2x + 1. We walked through the steps of simplifying both sides, moving terms around, isolating 'x', and, most importantly, verifying our solution. We found that x = −5/32 is indeed the value that makes the equation true. Solving equations like this might seem tricky at first, but with practice and a step-by-step approach, you can conquer any algebraic challenge. Keep up the great work, and remember, math can be fun!