Solving For X: A Step-by-Step Guide

Hey math whizzes and anyone who’s ever stared blankly at an equation, let’s talk about solving for x! It’s one of those fundamental skills in math that pops up everywhere, from your algebra class to trying to figure out the best deal at the store. Today, we’re going to dive deep into a specific type of equation, the kind that looks a little intimidating with fractions and variables hanging out in the denominator. You know the ones I’m talking about! We’re going to break down how to tackle an equation like $\frac{3}{x-2}-5=\frac{3(x-1)}{x-2}$ and emerge victorious. No fear, just pure problem-solving power! This isn't just about getting the right answer; it’s about understanding the why behind each step. We’ll walk through it together, dissecting the process so you can confidently approach similar problems. Get ready to level up your math game, guys!

Understanding the Equation: What Are We Dealing With?

Alright, let’s get down to business with our equation: $\frac{3}{x-2}-5=\frac{3(x-1)}{x-2}$. The first thing you probably notice is that dreaded denominator, $x-2$. This is super important because it tells us something crucial: $x$ cannot be equal to 2. If $x$ were 2, we’d be dividing by zero, and that’s a big no-no in math – it’s undefined! So, right off the bat, we establish our exclusion: $x \neq 2$. Keep this in the back of your mind; it’s our golden rule for this problem. Why is this so important? Because sometimes, when we solve equations, we might get an answer that violates this rule. If that happens, that particular solution is extraneous, meaning it’s not a valid solution to the original equation. So, always, always check your final answers against your initial exclusions. It’s like having a secret handshake to make sure your solution is legit.

Now, let’s look at the structure. We have fractions on both sides, and these fractions share the same denominator, $x-2$. This is a gift, folks! When denominators are the same, it makes things much, much simpler. It means we can clear out these fractions pretty easily. Our goal is to isolate $x$. To do that, we need to get rid of the denominators and combine like terms. Think of it like tidying up a messy room – you want everything in its place, and the variable x should be all by itself on one side of the equals sign. We’ve got a constant term (-5) and terms with x in the numerator of fractions. Our strategy will involve multiplying the entire equation by the common denominator to eliminate those pesky fractions, then simplifying and solving for x. It’s a systematic approach that breaks down a complex-looking problem into manageable steps. So, take a deep breath, identify those key features (like the exclusion), and let’s move on to the next step of actually solving it.

Step 1: Eliminate the Denominators

Okay, team, here’s where the magic happens! To get rid of those fractions in our equation $\frac{3}{x-2}-5=\frac{3(x-1)}{x-2}$, we’re going to multiply every single term on both sides of the equation by the common denominator, which is $x-2$. Remember our exclusion? $x \neq 2$. This multiplication step is only valid if $x-2$ is not zero. So, by multiplying by $x-2$, we’re essentially assuming $x \neq 2$, which aligns perfectly with our initial condition. Let's write it out:

$(x-2) \times \left(\frac{3}{x-2}-5\right) = (x-2) \times \left(\frac{3(x-1)}{x-2}\right)$

Now, we distribute the $(x-2)$ to each term on the left side:

$(x-2) \times \frac{3}{x-2} - (x-2) \times 5 = (x-2) \times \frac{3(x-1)}{x-2}$

Look at this beauty! On the left side, the $(x-2)$ in the first term cancels out with the denominator $(x-2)$. Boom! Fraction gone.

$3 - (x-2) \times 5 = (x-2) \times \frac{3(x-1)}{x-2}$

On the right side, the $(x-2)$ in the denominator also cancels out with the $(x-2)$ we are multiplying by. Poof! Another fraction bites the dust.

$3 - (x-2) \times 5 = 3(x-1)$

This is the goal, guys! By multiplying by the common denominator, we’ve successfully transformed our equation with fractions into a much simpler linear equation without any fractions. This is a huge leap forward in solving for x. It might seem like a small step, but it’s the most critical one for simplifying equations with rational expressions. Always be on the lookout for common denominators; they are your best friends in these situations. It streamlines the entire process and reduces the chance of making errors with fraction arithmetic. Now that we’ve cleared the decks of those fractions, we can move on to simplifying and solving the resulting equation.

Step 2: Simplify Both Sides of the Equation

We’ve banished the fractions, which is awesome! Our equation now looks like this: $3 - (x-2) \times 5 = 3(x-1)$. The next logical step is to simplify both sides of the equation as much as possible. Think of this as cleaning up the terms so we can see what we’re really working with. On the left side, we have $3 - (x-2) \times 5$. We need to distribute the -5 to both terms inside the parentheses: $-5 \times x$ and $-5 \times -2$.

So, $-(x-2) imes 5$ becomes $-5x + 10$.

Putting it all together on the left side, we get: $3 - 5x + 10$. We can combine the constant terms (3 and 10) to get $13 - 5x$. So, the left side simplifies to $13 - 5x$.

Now, let's look at the right side: $3(x-1)$. Here, we need to distribute the 3 to both terms inside the parentheses: $3 \times x$ and $3 \times -1$.

So, $3(x-1)$ becomes $3x - 3$.

Our equation, after simplification, now looks like this: $13 - 5x = 3x - 3$.

See how much cleaner that is? We’ve gone from an equation with fractions to a straightforward linear equation. This simplification step is crucial because it prepares the equation for the final steps of isolating x. It involves basic arithmetic and the distributive property, skills you’ve definitely honed. Always double-check your distribution and combining of like terms. A small error here can throw off your entire solution. It’s like making sure all your ingredients are correctly measured before you start baking – accuracy is key! Now that both sides are simplified, we’re one step closer to finding the value of x. Get ready for the final push to get x all by itself!

Step 3: Isolate the Variable (x)

We’re in the home stretch, guys! Our simplified equation is $13 - 5x = 3x - 3$. The mission now is to get all the x terms on one side of the equation and all the constant terms on the other. This is the classic