Solve 4x - 6 = 10x - 3: Step-by-Step Guide

Hey everyone! Today, we're diving into a classic algebra problem that might seem a little daunting at first glance, but trust me, it's totally manageable. We're going to tackle the equation $4x - 6 = 10x - 3$ and find out what the value of $x$ is. Whether you're a math whiz or just trying to get a handle on your algebra homework, this guide is for you. We'll break it down step-by-step, making sure you understand each part of the process. So, grab a pen and paper, and let's get started on solving this equation!

Understanding the Goal: Isolating 'x'

Alright guys, the main goal when we're faced with an equation like $4x - 6 = 10x - 3$ is to isolate the variable $x$. Think of it like a puzzle where you need to get $x$ all by itself on one side of the equals sign. To do this, we'll use a few basic algebraic rules that are super important. These rules essentially say that whatever you do to one side of the equation, you have to do the exact same thing to the other side. This keeps the equation balanced, like a perfectly calibrated scale. We'll be moving terms around, combining like terms, and ultimately figuring out the specific numerical value that $x$ represents to make this equation true.

It's crucial to remember that the equals sign isn't just a separator; it signifies balance. If you have $5$ on the left and $5$ on the right, they're equal. If you add $2$ to the left, making it $7$, you must add $2$ to the right, making it $7$ as well, to maintain that equality. We use this principle to move numbers and variables around. For example, if you have a term you want to get rid of on one side, say a '+5', you can subtract $5$ from that side. But remember, you must also subtract $5$ from the other side. This might seem like a lot of steps, but each one moves us closer to our final answer – the value of $x$. So, let's get our hands dirty and start applying these principles to our specific equation!

Step 1: Gathering 'x' Terms

Okay, so the first tactical move in solving $4x - 6 = 10x - 3$ is to get all the terms containing $x$ onto one side of the equation. It doesn't really matter which side you choose, but often it's easier to move the smaller $x$ term to the side with the larger $x$ term. This helps avoid dealing with negative coefficients for $x$ right away, though it's not strictly necessary. In our case, we have $4x$ on the left and $10x$ on the right. Since $10x$ is bigger than $4x$, let's move the $4x$ over to the right side.

To move the $4x$ from the left side, we need to perform the opposite operation. Since it's currently $4x$ (which is $4x$ plus nothing else on that side), we'll subtract $4x$. But remember the golden rule: whatever we do to one side, we must do to the other. So, we subtract $4x$ from both sides of the equation:

$4x - 6 - 4x = 10x - 3 - 4x$

Now, let's simplify both sides. On the left side, $4x - 4x$ cancels out, leaving us with just $-6$. On the right side, $10x - 4x$ gives us $6x$. So, our equation now looks much cleaner:

$-6 = 6x - 3$

See? We've successfully gathered all the $x$ terms onto one side. This is a huge step towards finding our solution!

Step 2: Isolating the 'x' Term

Now that we have our $x$ terms consolidated on one side (the right side, in this case, giving us $6x$), our next mission is to isolate this $6x$ term. This means we need to get rid of any other numbers that are hanging out with it. Looking at our current equation, $-6 = 6x - 3$, we see that the $6x$ term has a $-3$ next to it.

To get $6x$ by itself, we need to undo the subtraction of $3$. The opposite of subtracting $3$ is adding $3$. So, we're going to add $3$ to both sides of the equation to keep things balanced:

$-6 + 3 = 6x - 3 + 3$

Let's simplify again. On the left side, $-6 + 3$ equals $-3$. On the right side, $-3 + 3$ cancels out, leaving us with just $6x$. Our equation is now:

$-3 = 6x$

We are so close, guys! The $6x$ term is almost isolated. We just have one more step to get $x$ completely by itself.

Step 3: Solving for 'x'

The final frontier in solving $-3 = 6x$ is to get $x$ completely alone. Right now, $x$ is being multiplied by $6$. To reverse multiplication, we use division. So, we need to divide both sides of the equation by $6$.

rac{-3}{6} = rac{6x}{6}

On the left side, rac{-3}{6} simplifies. Both $3$ and $6$ are divisible by $3$. So, -3 rf 3 = -1 and 6 rf 3 = 2. This gives us -rac{1}{2}.

On the right side, rac{6x}{6} simplifies to just $x$, because the $6$'s cancel out.

So, our final answer is:

x = -rac{1}{2}

And there you have it! We've successfully solved the equation.

Step 4: Checking Your Answer (The Cool Down Lap)

Now, a really smart move in algebra, especially when you're learning, is to check your answer. This is like a cool-down lap after a race – it makes sure you didn't miss any steps or make a silly calculation error. It gives you confidence that your solution is indeed correct.

Our solution is x = -rac{1}{2}. To check it, we substitute this value back into the original equation: $4x - 6 = 10x - 3$. Let's see if both sides end up being equal.

First, let's evaluate the left side (LS): $4x - 6$

Substitute x = -rac{1}{2}:

LS = 4 imes (-rac{1}{2}) - 6

LS = -rac{4}{2} - 6

LS = $-2 - 6$

LS = $-8$

Now, let's evaluate the right side (RS): $10x - 3$

Substitute x = -rac{1}{2}:

RS = 10 imes (-rac{1}{2}) - 3

RS = -rac{10}{2} - 3

RS = $-5 - 3$

RS = $-8$

Look at that! The left side equals $-8$ and the right side also equals $-8$. Since LS = RS ($-8 = -8$), our solution x = -rac{1}{2} is absolutely correct! This verification step is super valuable for confirming your work.

Conclusion: You've Got This!

So there you have it, guys! We took the equation $4x - 6 = 10x - 3$ and, by following a clear, step-by-step process, we arrived at the solution x = -rac{1}{2}. We gathered the $x$ terms, isolated the $x$ term, and finally solved for $x$. And to top it off, we even checked our answer to make sure it was spot on. Algebra might seem tricky sometimes, but with a little practice and by understanding the core principles of balancing equations, you can tackle any problem thrown your way. Keep practicing, and you'll become a pro in no time!

The key takeaways here are to always perform the same operation on both sides of the equation and to use inverse operations to isolate your variable. Whether it's adding to subtract, multiplying to divide, or vice versa, these are your tools for success. Remember that checking your answer isn't just a suggestion; it's a powerful way to build confidence in your mathematical abilities. So next time you're faced with a linear equation, remember these steps, stay calm, and solve it systematically. You've totally got this!