Solving 7 + X - 15 = -2 ⅔ A Step-by-Step Guide
Hey guys! Let's dive into solving this equation: 7 + x - 15 = -2 ⅔. This might seem a bit intimidating at first, especially with that fraction thrown in, but trust me, it's totally manageable. We'll break it down step by step, so you'll be solving equations like a pro in no time. Understanding how to solve for x in equations like this is a foundational skill in algebra. It pops up everywhere, from simple word problems to more complex mathematical models. So, let's get this nailed down, shall we?
Understanding the Basics of Algebraic Equations
Before we jump into the specific equation, let’s quickly recap some fundamental concepts. An algebraic equation is essentially a mathematical statement that shows the equality between two expressions. Our goal when solving for a variable, like x in our case, is to isolate that variable on one side of the equation. Think of it like balancing a scale: whatever you do to one side, you must do to the other to keep things balanced. This principle is crucial for maintaining the equation's integrity and arriving at the correct solution. The main tools we use for this balancing act are inverse operations. Addition and subtraction are inverse operations, as are multiplication and division. By applying these operations strategically, we can effectively peel away the layers surrounding our variable until it stands alone, revealing its value. Remember, the key is consistency. If you add 5 to the left side, you've got to add 5 to the right side. No exceptions! This meticulous approach ensures that the equation remains balanced and your solution remains valid. We often use the properties of equality, such as the addition property (adding the same value to both sides) and the subtraction property (subtracting the same value from both sides), to manipulate equations and move closer to our goal. So, with these basics in mind, let's tackle our equation head-on and see how we can put these principles into action.
Step 1: Simplify the Left Side of the Equation
Okay, so our equation is 7 + x - 15 = -2 ⅔. The first thing we want to do is simplify things on the left side. We've got some constants – that's the 7 and the -15 – that we can combine. This is where our basic arithmetic skills come into play. Think of it as simply adding and subtracting numbers. So, we have 7 minus 15. What does that give us? That's right, it's -8. So now, we can rewrite the left side of our equation. Instead of 7 + x - 15, we now have x - 8. See how much cleaner that looks already? This step might seem small, but it's super important. By combining those constants, we've reduced the clutter and made the equation easier to work with. It's all about making things manageable. Remember, in mathematics, simplifying expressions is often the first step towards solving a problem. It's like decluttering your workspace before you start a project – it just makes everything smoother. This simplified form, x - 8, is much easier to handle when we move on to the next step of isolating x. So, let's keep this momentum going and see what we need to do next to get that x all by itself on one side of the equation.
Step 2: Convert the Mixed Number to an Improper Fraction
Now, let's tackle the right side of our equation: -2 ⅔. We've got a mixed number here, and mixed numbers can sometimes be a bit tricky to work with directly, especially when we're dealing with equations. So, the best thing to do is convert it into an improper fraction. An improper fraction is simply a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert -2 ⅔ to an improper fraction, we're going to follow a simple process. First, we multiply the whole number part (which is 2) by the denominator (which is 3). That gives us 2 * 3 = 6. Then, we add that result to the numerator (which is 2). So, 6 + 2 = 8. This 8 becomes our new numerator. We keep the same denominator, which is 3. And don't forget the negative sign! So, -2 ⅔ becomes -8/3. This conversion is super helpful because improper fractions are much easier to work with when we're performing operations like addition, subtraction, multiplication, and division. Now that we've transformed -2 ⅔ into -8/3, our equation is starting to look even more manageable. We're one step closer to isolating x and finding our solution. This step is a common practice in algebra, so mastering it will definitely pay off as you tackle more complex equations.
Step 3: Isolate x by Adding 8 to Both Sides
Alright, we're making some serious progress! Our equation now looks like x - 8 = -8/3. Our mission is to get that x all by itself on the left side. To do that, we need to get rid of the -8 that's hanging out with it. Remember our balancing act? Whatever we do to one side, we have to do to the other. So, to cancel out the -8, we're going to add 8 to both sides of the equation. This is the magic of inverse operations in action! When we add 8 to the left side (x - 8 + 8), the -8 and the +8 cancel each other out, leaving us with just x. Perfect! But we can't forget the right side. We need to add 8 to -8/3 as well. So, we have -8/3 + 8. Now, we're dealing with fractions again, but don't worry, we've got this. To add these together, we need a common denominator. We can think of 8 as 8/1. The common denominator for 3 and 1 is 3. So, we need to convert 8/1 into a fraction with a denominator of 3. To do that, we multiply both the numerator and denominator by 3: 8/1 * (3/3) = 24/3. Now we can add: -8/3 + 24/3. This gives us 16/3. So, after adding 8 to both sides, our equation becomes x = 16/3. We're almost there! We've successfully isolated x, but let's take one more step to make our answer even clearer.
Step 4: Convert the Improper Fraction Back to a Mixed Number (Optional)
We've arrived at x = 16/3. This is a perfectly valid answer, but sometimes it's helpful to express it as a mixed number, especially if we're dealing with real-world applications where mixed numbers might make more intuitive sense. So, let's convert 16/3 back into a mixed number. To do this, we're going to divide the numerator (16) by the denominator (3). How many times does 3 go into 16? It goes in 5 times, because 5 * 3 = 15. That means 5 is our whole number part. But we're not quite done yet. We have a remainder. 16 minus 15 is 1, so our remainder is 1. This remainder becomes the numerator of our fractional part, and we keep the same denominator, which is 3. So, 16/3 is equal to 5 ⅓. Therefore, we can also express our solution as x = 5 ⅓. Whether you leave your answer as 16/3 or convert it to 5 ⅓ is often a matter of preference or the specific instructions of a problem. Both forms represent the same value, but the mixed number might be easier to visualize in some contexts. And there you have it! We've successfully solved for x. Let's just do a quick recap of all the steps we took.
Final Answer: x = 16/3 or x = 5 ⅓
So, just to recap, we started with the equation 7 + x - 15 = -2 ⅔. We simplified the left side, converted the mixed number to an improper fraction, isolated x by adding 8 to both sides, and then, optionally, we converted the improper fraction back to a mixed number. We found that x = 16/3, which is the same as x = 5 ⅓. You guys nailed it! See, solving equations isn't so scary when you break it down step by step. By simplifying, using inverse operations, and keeping the equation balanced, you can tackle all sorts of algebraic challenges. Remember, practice makes perfect, so the more equations you solve, the more confident you'll become. Keep up the great work, and you'll be a math whiz in no time! If you ever get stuck, just remember these steps and take it one piece at a time. You've got this!
