Solving The Equation: X + 2X + 5X + 50 + 40 - 10 = 360

Hey guys! Let's dive into solving this equation together. It looks a bit intimidating at first, but don't worry, we'll break it down step by step. Our main goal here is to find the value of 'X' that makes the equation true. So, grab your thinking caps, and let's get started!

Understanding the Equation

The equation we're tackling is: X + 2X + 5X + 50 + 40 - 10 = 360. Before we jump into solving it, let's understand what each part means. We have terms involving 'X', which are variables, and we have constant numbers. Our mission is to simplify and isolate 'X' on one side of the equation.

Combining Like Terms

The first thing we need to do is to combine the like terms. Like terms are those that have the same variable (in this case, 'X') or are just constant numbers. Let's start by combining the 'X' terms: X + 2X + 5X. Think of it as adding apples. If you have 1 apple (X), and you add 2 more apples (2X), and then 5 more apples (5X), how many apples do you have in total? You'd have 8 apples (8X). So, X + 2X + 5X simplifies to 8X. This is a crucial step because it reduces the complexity of the equation significantly. By combining these terms, we make the equation easier to work with. It’s like gathering all similar items together before sorting them. This helps in organizing our approach and prevents us from getting overwhelmed by too many individual terms.

Simplifying the Constants

Next, let's combine the constant numbers: 50 + 40 - 10. This part is just basic arithmetic. We add 50 and 40 to get 90, and then we subtract 10 from 90. So, 50 + 40 - 10 equals 80. Dealing with the constants in this way helps to streamline the equation, making it clearer and more manageable. Just as combining like variables simplifies the variable side, simplifying constants makes the numerical side of the equation less cluttered. This is essential for accurately solving the equation, as each simplification brings us closer to isolating 'X'. Think of it like decluttering your workspace before starting a project; a clean and organized space makes the task at hand much easier to handle.

Isolating the Variable

Now that we've simplified the equation, let's rewrite it with our combined terms: 8X + 80 = 360. The next key step is to isolate the term with 'X' on one side of the equation. This means we want to get '8X' by itself. To do this, we need to get rid of the '+ 80' on the left side. Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. It's like a seesaw; if you add weight to one side, you need to add the same weight to the other side to keep it level. The operation that undoes addition is subtraction. So, to remove the '+ 80', we'll subtract 80 from both sides of the equation.

Subtracting from Both Sides

Subtracting 80 from both sides gives us: 8X + 80 - 80 = 360 - 80. On the left side, the '+ 80' and '- 80' cancel each other out, leaving us with just '8X'. On the right side, 360 minus 80 equals 280. So, our equation now looks like this: 8X = 280. We're getting closer! This step is crucial because it brings us one step closer to isolating 'X'. By performing the same operation on both sides, we maintain the balance of the equation while simplifying it. It’s like peeling away layers of an onion; each layer removed brings us closer to the core.

Dividing to Solve for X

We now have 8X = 280. To finally solve for 'X', we need to isolate 'X' completely. Currently, 'X' is being multiplied by 8. To undo this multiplication, we need to perform the opposite operation, which is division. Just as subtraction undoes addition, division undoes multiplication. We'll divide both sides of the equation by 8. This keeps the equation balanced, which is essential for finding the correct solution. It's like ensuring that both sides of a scale are equal before reading the measurement. Dividing both sides by the same number ensures that the value of 'X' we find is accurate and valid for the original equation.

The Final Solution

Dividing both sides by 8, we get: 8X / 8 = 280 / 8. On the left side, 8X divided by 8 simplifies to just 'X'. On the right side, 280 divided by 8 equals 35. So, our final solution is: X = 35. Hooray! We've solved the equation. This means that the value of 'X' that makes the original equation true is 35. It's like finding the missing piece of a puzzle; once we know the value of 'X', we can plug it back into the original equation to check our work and ensure that both sides balance.

Checking Our Work

It's always a good idea to check our solution to make sure we didn't make any mistakes along the way. To do this, we'll substitute 'X = 35' back into the original equation: 35 + 2(35) + 5(35) + 50 + 40 - 10 = 360. Let's simplify this: 35 + 70 + 175 + 50 + 40 - 10. Now, let's add all the numbers together: 35 + 70 = 105, 105 + 175 = 280, 280 + 50 = 330, 330 + 40 = 370, and finally, 370 - 10 = 360. So, we get 360 = 360, which confirms that our solution is correct. Checking our work is like proofreading a document before submitting it; it helps us catch any errors and ensures that our answer is accurate.

Conclusion

So, there you have it! We've successfully solved the equation X + 2X + 5X + 50 + 40 - 10 = 360, and we found that X = 35. Remember, the key to solving equations is to simplify, combine like terms, and isolate the variable. Keep practicing, and you'll become a pro at solving equations in no time. Great job, everyone!