Solving For X: -9/x = 6 - A Step-by-Step Guide
Hey guys! Ever get those equations that look a little intimidating at first glance? Well, don't sweat it! We're going to break down one of those today: solving for x in the equation -9/x = 6. This might seem tricky, but I promise, with a few simple steps, you'll be a pro at solving these types of problems. We'll go through each step in detail, making sure you understand the why behind the how. So, grab your pencils, and let's dive in!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation is telling us. The equation -9/x = 6 basically says that if you divide -9 by some number (x), the result will be 6. Our mission, should we choose to accept it (and we do!), is to figure out what that mystery number x is.
Now, you might be tempted to guess and check, but in mathematics, we like to be a bit more systematic. That's where algebra comes in! We're going to use algebraic techniques to isolate x on one side of the equation, which will reveal its value. Remember, the golden rule of algebra is that whatever you do to one side of the equation, you must do to the other. This keeps the equation balanced and true.
Think of it like a scale: if you add weight to one side, you need to add the same weight to the other to keep it level. Equations are the same way! So, with that in mind, let's get started on the actual solving process.
Step 1: Getting Rid of the Fraction
Fractions can sometimes look a little scary, but they're really not that bad. Our first goal is to get rid of the fraction in our equation, -9/x = 6. To do this, we're going to multiply both sides of the equation by x. Why x? Because multiplying -9/x by x will cancel out the x in the denominator. It's like magic, but it's actually just math!
So, let's do it. We multiply both sides of the equation by x:
x * (-9/x) = 6 * x
On the left side, the x in the numerator and the x in the denominator cancel each other out, leaving us with just -9. On the right side, we simply have 6x. Our equation now looks like this:
-9 = 6x
See? We're already making progress! The fraction is gone, and things are looking much simpler. Now we have a more straightforward equation to work with. This step is crucial because it simplifies the equation and makes it easier to isolate our variable, x. Next up, we'll isolate that x and find its value.
Step 2: Isolating x
Great job on getting rid of the fraction! Now, let's isolate x. Our equation currently looks like this: -9 = 6x. Remember, isolating x means getting it all by itself on one side of the equation. Right now, x is being multiplied by 6. So, to undo this multiplication, we need to do the opposite operation: division.
We're going to divide both sides of the equation by 6. This will cancel out the 6 on the right side, leaving x alone. Let's do it:
-9 / 6 = (6x) / 6
On the right side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just x. On the left side, we have -9 / 6. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 3. This gives us -3/2. Our equation now looks like this:
-3/2 = x
Or, if we flip it around to the more conventional way of writing the solution, we have:
x = -3/2
Boom! We've done it! We've isolated x and found its value. x is equal to -3/2. That wasn't so bad, was it? Just a couple of steps and we cracked the code. Now, let's just double-check our work to make sure we got it right.
Step 3: Checking Our Answer
It's always a good idea to check your answer, especially in math. This helps you catch any mistakes and ensures that your solution is correct. To check our answer, we're going to plug our value for x (which is -3/2) back into the original equation, -9/x = 6, and see if it holds true.
So, let's substitute -3/2 for x:
-9 / (-3/2) = 6
Dividing by a fraction can be a little tricky, but remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of -3/2 is -2/3. So, we can rewrite the equation as:
-9 * (-2/3) = 6
Now, let's multiply. A negative times a negative is a positive, so we have:
(9 * 2) / 3 = 6
18 / 3 = 6
6 = 6
Woohoo! It checks out! The left side of the equation equals the right side, which means our solution, x = -3/2, is correct. We did it! Checking our answer gives us confidence that we've solved the problem accurately. Plus, it's a good habit to get into for any math problem.
Wrapping Up
So there you have it, guys! We've successfully solved for x in the equation -9/x = 6. We went through each step carefully, from getting rid of the fraction to isolating x and finally checking our answer. Remember, the key to solving algebraic equations is to understand the underlying principles and to take things one step at a time. Don't be afraid of fractions or negative numbers – they're just part of the mathematical landscape!
Key takeaways from this problem:
- Fractions: To eliminate a variable in the denominator, multiply both sides of the equation by that variable.
- Isolating Variables: To isolate a variable, perform the opposite operation (addition/subtraction, multiplication/division) on both sides of the equation.
- Checking Your Work: Always plug your solution back into the original equation to verify its correctness.
Practice makes perfect, so try solving similar equations on your own. The more you practice, the more comfortable you'll become with these techniques. And remember, math can be fun! Keep exploring, keep learning, and keep solving! You've got this!
