Solving The Equation: -7/5 ÷ X * 8 = 150 - 100

Hey guys! Today, we're diving into a fun math problem. We're going to break down and solve the equation -7/5 ÷ x * 8 = 150 - 100. Math can sometimes seem intimidating, but don't worry, we'll take it step by step. Our main goal here is to figure out the value of 'x' that makes this equation true. Think of it like a puzzle where we need to find the missing piece. So, grab your pencils and let's get started! We’ll be using basic algebraic principles to isolate ‘x’ and find its value. Remember, math is all about understanding the steps, not just memorizing them, so let’s make sure we grasp each one as we go. Let's make math fun and conquer this equation together!

Understanding the Equation

First, let's break down the equation -7/5 ÷ x * 8 = 150 - 100 piece by piece. This will help us understand what we're dealing with and how to approach solving it. We have fractions, division, multiplication, subtraction, and our unknown variable 'x'. It might seem like a lot, but we can handle it! The left side of the equation involves dividing -7/5 by 'x' and then multiplying the result by 8. The right side is a simple subtraction: 150 minus 100. Understanding the order of operations (PEMDAS/BODMAS) is crucial here. We need to perform operations in the correct sequence to get the right answer. PEMDAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures we solve the equation in a logical manner. For example, we'll deal with the division and multiplication before we even think about any addition or subtraction that might pop up later. So, let’s keep PEMDAS in mind as we move forward. By carefully dissecting the equation, we can create a clear roadmap for solving it. Remember, a little bit of understanding goes a long way in math!

Step 1: Simplify the Right Side

Let's start by simplifying the right side of the equation: 150 - 100. This is a straightforward subtraction, which makes our task a bit easier right off the bat. When we subtract 100 from 150, we get 50. So, now our equation looks like this: -7/5 ÷ x * 8 = 50. See? We've already made progress! Simplifying one side of the equation helps to make the whole problem less cluttered and easier to manage. This is a common strategy in algebra – break things down into smaller, more digestible parts. By reducing the complexity on one side, we can focus our attention on the remaining operations and make sure we handle them correctly. Think of it as decluttering your workspace before starting a project; a cleaner equation is much easier to work with. Plus, getting a quick win like this can give you a nice boost of confidence to tackle the rest of the problem. So, with the right side simplified, we're ready to move on to the next step and continue our journey towards finding the value of 'x'.

Step 2: Rewrite the Division as Multiplication

Now, let's tackle the left side of the equation: -7/5 ÷ x * 8. The first thing we'll do is rewrite the division as multiplication. Remember, dividing by a number is the same as multiplying by its reciprocal. This is a key concept in algebra that helps simplify equations. So, instead of dividing by 'x', we'll multiply by 1/x. This transforms our equation segment from -7/5 ÷ x to -7/5 * (1/x). Now our equation looks like this: -7/5 * (1/x) * 8 = 50. This change might seem small, but it makes a big difference in how we can manipulate the equation. Multiplication is often easier to work with than division, especially when we're dealing with fractions and variables. By rewriting the division, we've set ourselves up for the next step, which will involve combining these terms and getting closer to isolating 'x'. Think of it as converting to a common language – now all our operations on the left side are speaking the same "multiplication" language, making it easier to manage and solve. So, with this division-to-multiplication trick up our sleeves, we're well-prepared to move forward.

Step 3: Combine the Terms on the Left Side

Okay, let's keep the momentum going! We've rewritten our equation to -7/5 * (1/x) * 8 = 50. Now it's time to combine the terms on the left side. We have a series of multiplications here, which we can perform in any order. Let's start by multiplying -7/5 by 8. To do this, we multiply the numerator (-7) by 8, which gives us -56. The denominator remains 5. So, -7/5 * 8 becomes -56/5. Now our equation looks like this: -56/5 * (1/x) = 50. We're getting closer! The next step is to multiply -56/5 by 1/x. This simply means multiplying the numerators and the denominators. So, -56/5 * (1/x) becomes -56/(5x). This gives us a single fraction on the left side, which is much easier to handle. Now our equation is: -56/(5x) = 50. By combining these terms, we've simplified the left side into a more manageable expression. Think of it as condensing all the ingredients into a single mixture – now we have a unified expression that we can work with. This makes the next steps in solving for 'x' much clearer and more straightforward. So, let's take a deep breath and get ready to isolate 'x'!

Step 4: Isolate x

Alright, we're in the home stretch! Our equation now looks like -56/(5x) = 50. Our main goal here is to isolate 'x', which means getting 'x' by itself on one side of the equation. To do this, we'll first get rid of the fraction. We can do this by multiplying both sides of the equation by 5x. This is a crucial step because it eliminates the denominator and simplifies our equation. When we multiply both sides by 5x, we get: (-56/(5x)) * 5x = 50 * 5x. On the left side, the 5x in the numerator and denominator cancel each other out, leaving us with -56. On the right side, 50 * 5x becomes 250x. So, our equation now looks like this: -56 = 250x. We're almost there! Now, to completely isolate 'x', we need to divide both sides of the equation by 250. This will undo the multiplication and leave 'x' by itself. So, we divide both sides by 250: -56 / 250 = (250x) / 250. The 250s on the right side cancel out, leaving us with 'x'. So, we have: x = -56 / 250. By performing these steps, we've successfully isolated 'x' and found a value for it. This is the heart of solving any equation – strategically manipulating it until you get the variable alone. Now, let’s simplify this fraction to get our final answer.

Step 5: Simplify the Fraction

We've found that x = -56 / 250. Now, let's simplify this fraction to its simplest form. Simplifying fractions makes the answer cleaner and easier to understand. To simplify, we need to find the greatest common divisor (GCD) of the numerator (-56) and the denominator (250). The GCD is the largest number that divides both numbers evenly. The GCD of 56 and 250 is 2. So, we'll divide both the numerator and the denominator by 2. When we divide -56 by 2, we get -28. When we divide 250 by 2, we get 125. So, our simplified fraction is x = -28 / 125. This is the simplest form of the fraction, and it represents the final value of 'x' that solves our equation. Simplifying the fraction is like putting the final touches on a masterpiece – it makes the answer polished and clear. We've taken a potentially messy fraction and turned it into something neat and tidy. So, with this simplified value of 'x', we've officially solved the equation! Let’s give ourselves a pat on the back for sticking with it and cracking the code.

Final Answer

So, after all our hard work, we've arrived at the final answer! We started with the equation -7/5 ÷ x * 8 = 150 - 100 and, step by step, we've solved for 'x'. We simplified the right side, rewrote division as multiplication, combined terms, isolated 'x', and simplified the resulting fraction. Our final answer is x = -28 / 125. Woohoo! We did it! This value of 'x' is the solution to the equation, meaning that if we plug -28/125 back into the original equation in place of 'x', the equation will hold true. It’s always a good idea to double-check your work, so you might want to try plugging this value back into the original equation to verify that it works. Solving equations like this is a fundamental skill in algebra and mathematics. It’s like having a superpower that allows you to unlock mathematical puzzles. By mastering these steps, you’ll be well-equipped to tackle more complex equations and mathematical challenges in the future. So, congratulations on solving this equation with me, and keep up the great work!