Solving For X: A Step-by-Step Guide
Hey guys! Today, we're diving into a classic algebra problem: solving for x in an equation involving fractions. Specifically, we're going to tackle the equation (x-7)/(x-9) = 3/4. Don't worry if this looks intimidating at first; we'll break it down into easy-to-follow steps. So, grab your pencils and let's get started!
Understanding the Equation
Before we jump into the solution, let's take a moment to understand what the equation (x-7)/(x-9) = 3/4 actually means. This equation states that the ratio of (x-7) to (x-9) is equal to the ratio of 3 to 4. In simpler terms, if we plug in the correct value for x, the left side of the equation will equal the right side. Our goal is to find that specific value of x that makes this equation true.
The key here is recognizing that we're dealing with fractions and proportions. This means we can use some neat tricks to manipulate the equation and isolate x. One of the most important techniques we'll use is cross-multiplication, but before we get there, let's quickly touch on why certain values of x might be problematic. Specifically, we need to be mindful of the denominator, (x-9). If x were to equal 9, the denominator would become zero, and we all know division by zero is a big no-no in the math world! It leads to undefined results and can throw off our entire solution. So, we'll keep this in mind as we work through the problem and make sure our final answer doesn't make the denominator zero.
Why is this important? Well, think of it like this: you can't divide a pizza into zero slices! It just doesn't make sense. Similarly, in math, dividing by zero creates mathematical chaos. So, by being aware of this potential issue, we're setting ourselves up for a smooth and accurate solution. We're like math detectives, carefully looking for clues and avoiding pitfalls along the way. So, let's keep this in the back of our minds as we move forward and prepare to solve for x!
Step 1: Cross-Multiplication
The first and often most crucial step in solving an equation like (x-7)/(x-9) = 3/4 is cross-multiplication. This technique allows us to get rid of the fractions and transform the equation into a more manageable form. But what exactly is cross-multiplication, and why does it work?
Cross-multiplication is essentially a shortcut for multiplying both sides of the equation by the denominators. Think of it like this: to eliminate the fraction on the left side, we could multiply both sides of the equation by (x-9). Similarly, to eliminate the fraction on the right side, we could multiply both sides by 4. Cross-multiplication combines these two steps into one neat process.
Here's how it works in practice:
- Multiply the numerator of the left fraction (x-7) by the denominator of the right fraction (4): 4 * (x-7).
- Multiply the numerator of the right fraction (3) by the denominator of the left fraction (x-9): 3 * (x-9).
- Set these two products equal to each other: 4(x-7) = 3(x-9).
So, by cross-multiplying, we've transformed our original equation (x-7)/(x-9) = 3/4 into a linear equation: 4(x-7) = 3(x-9). This is a significant step forward because we've eliminated the fractions, making the equation much easier to solve. But why does this work? It all comes down to the fundamental properties of equality. When we perform the same operation on both sides of an equation, we maintain the balance. In this case, we're effectively multiplying both sides by the common denominator, which clears the fractions without changing the solution.
Now that we've successfully cross-multiplied, we're ready to move on to the next step: simplifying the equation. This involves distributing, combining like terms, and isolating x. So, let's keep that momentum going and see how we can further unravel this equation!
Step 2: Simplify the Equation
Alright, guys, we've successfully cross-multiplied and arrived at the equation 4(x-7) = 3(x-9). Now it's time to simplify things and get closer to solving for x. This involves a couple of key steps: distributing and combining like terms. Let's break it down.
First up, distribution. We need to get rid of those parentheses by multiplying the numbers outside the parentheses by each term inside. On the left side, we have 4(x-7). This means we need to multiply 4 by both x and -7. So, 4 * x = 4x, and 4 * -7 = -28. This gives us 4x - 28.
On the right side, we have 3(x-9). Similarly, we multiply 3 by both x and -9. So, 3 * x = 3x, and 3 * -9 = -27. This gives us 3x - 27.
Now our equation looks like this: 4x - 28 = 3x - 27. Much cleaner, right? We've eliminated the parentheses and have a linear equation with x terms and constant terms on both sides. The next step is to combine like terms. This means grouping the x terms together and the constant terms together.
To do this, we want to get all the x terms on one side of the equation and all the constant terms on the other side. A common strategy is to move the smaller x term to the side with the larger x term. In this case, 3x is smaller than 4x, so let's subtract 3x from both sides of the equation. This gives us:
4x - 3x - 28 = 3x - 3x - 27 x - 28 = -27
Now we have x terms on the left and constant terms on the right, except for that pesky -28 on the left side. To isolate x completely, we need to get rid of that -28. We can do this by adding 28 to both sides of the equation:
x - 28 + 28 = -27 + 28 x = 1
And there you have it! We've successfully simplified the equation and isolated x. It looks like our solution is x = 1. But before we celebrate, there's one crucial step left: verifying our solution. We need to make sure that x = 1 actually works in the original equation and doesn't lead to any mathematical problems.
Step 3: Verify the Solution
Okay, we've arrived at a potential solution: x = 1. But in mathematics, it's always crucial to verify your solution. Think of it like double-checking your work on a test – you want to make sure you haven't made any mistakes along the way. In this case, verifying our solution means plugging x = 1 back into the original equation and seeing if it holds true.
Our original equation was (x-7)/(x-9) = 3/4. Let's substitute x with 1:
(1-7)/(1-9) = 3/4
Now, let's simplify the left side of the equation:
(-6)/(-8) = 3/4
We can further simplify the fraction -6/-8 by dividing both the numerator and denominator by their greatest common divisor, which is 2. Remember, a negative divided by a negative is a positive:
3/4 = 3/4
Look at that! The left side of the equation equals the right side of the equation. This confirms that x = 1 is indeed a valid solution. We've successfully verified our answer!
But there's another important reason why verification is essential in this type of problem. Remember earlier when we talked about the denominator (x-9)? We mentioned that x cannot equal 9 because that would make the denominator zero, leading to an undefined expression. By verifying our solution, we also make sure that our answer doesn't cause any such issues. In this case, x = 1 doesn't make the denominator zero, so we're in the clear.
Why is this so important? Imagine if we had arrived at a solution that made the denominator zero. We might think we had solved the problem, but our answer would actually be meaningless! Verification helps us catch these kinds of errors and ensures that our solution is both mathematically correct and logically sound.
So, we've not only found a solution, but we've also proven that it's the correct solution. That's a great feeling, right? Now, let's recap the entire process and solidify our understanding.
Conclusion
Alright, guys, we've reached the end of our journey to solve for x in the equation (x-7)/(x-9) = 3/4. Let's take a moment to recap the steps we took and celebrate our success!
We started by understanding the equation and recognizing that we were dealing with fractions and proportions. We also highlighted the importance of being mindful of the denominator and avoiding division by zero. This initial understanding set the stage for a smooth and accurate solution.
Next, we tackled the equation head-on using cross-multiplication. This powerful technique allowed us to eliminate the fractions and transform the equation into a more manageable linear form: 4(x-7) = 3(x-9). We discussed why cross-multiplication works, emphasizing the fundamental properties of equality.
With the fractions out of the way, we moved on to simplifying the equation. This involved distributing, combining like terms, and isolating x. We carefully worked through each step, showing how to get rid of parentheses and group similar terms together. This led us to a potential solution: x = 1.
But we didn't stop there! We knew that verification is crucial, so we verified our solution by plugging x = 1 back into the original equation. We showed that the left side of the equation equaled the right side, confirming that x = 1 is indeed the correct solution. We also reiterated the importance of verification in preventing errors and ensuring the validity of our answer.
So, what have we learned today? We've not only learned how to solve a specific equation, but we've also reinforced some fundamental algebraic principles. We've seen the power of cross-multiplication, the importance of simplification, and the necessity of verification. These are valuable skills that will serve you well in future math endeavors.
Solving equations like this can seem challenging at first, but with practice and a systematic approach, you can conquer any algebraic problem. Remember to break down the problem into smaller steps, stay organized, and always verify your solution. And most importantly, don't be afraid to ask for help when you need it!
So, keep practicing, keep exploring, and keep those math skills sharp. You've got this! Until next time, happy solving!