Solving For X: 3x - 5 = 10 - A Step-by-Step Guide

Hey guys! Let's dive into a super common algebra problem: solving for x in the equation 3x - 5 = 10. Don't worry, it's way easier than it looks. We'll break it down step by step so everyone can follow along. Think of it like following a recipe, but instead of cookies, we're baking up the value of x! So, grab your metaphorical aprons, and let’s get started!

Understanding Linear Equations

Before we jump into solving our specific equation, let's quickly recap what a linear equation actually is. A linear equation is basically an equation where the highest power of the variable (in our case, x) is 1. That means we won't see any x², x³, or anything like that. These equations, when graphed, form a straight line – hence the name “linear.”

Linear equations are fundamental in algebra and show up everywhere, from simple word problems to more complex scientific calculations. Being comfortable solving them is a crucial skill. The goal is always the same: to isolate the variable we're trying to find (that's x in our case) on one side of the equation. We do this by performing the same operations on both sides of the equation to maintain the balance.

Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. If you add 5 to one side, you need to add 5 to the other. If you multiply one side by 2, you need to multiply the other side by 2. This principle is the key to solving any algebraic equation, including our 3x - 5 = 10.

Why is understanding this important? Because it gives you the why behind the how. Instead of just memorizing steps, you understand the underlying principle that governs how we manipulate equations. This deeper understanding makes you a more confident and capable problem-solver. Plus, it allows you to tackle variations of the problem without getting tripped up. You'll be able to see the structure of the equation and know exactly what moves to make to isolate x.

Step 1: Isolating the Term with 'x'

Okay, so back to our equation: 3x - 5 = 10. Our first goal is to get the term with x (that's 3x) by itself on one side of the equation. To do this, we need to get rid of that pesky -5 that's hanging out with it. How do we do that? By performing the opposite operation!

Since we have “-5”, we need to add 5 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep things balanced. So, let's do it:

3x - 5 + 5 = 10 + 5

Notice what happens on the left side. The -5 and +5 cancel each other out, leaving us with just 3x:

3x = 15

Awesome! We've successfully isolated the term with x. We're one step closer to finding the value of x itself. This step is crucial because it simplifies the equation, making it easier to solve for the variable. By adding 5 to both sides, we've effectively moved the constant term to the right side of the equation, grouping all the constant terms together.

Think of it like sorting your laundry. You wouldn't wash your whites with your colors, right? Similarly, we want to group like terms together in our equation. By adding 5 to both sides, we're grouping the constant terms (the numbers without any variables) on the right side, making it easier to isolate and solve for x. So, remember this trick: to get rid of a term that's being added or subtracted, do the opposite operation to both sides of the equation!

Step 2: Solving for 'x'

Alright, we're at the home stretch! We've got 3x = 15. Now we need to get x all by itself. Right now, x is being multiplied by 3. So, to undo that multiplication, we need to do the opposite operation: division.

We're going to divide both sides of the equation by 3:

(3x) / 3 = 15 / 3

On the left side, the 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with just x:

x = 15 / 3

And finally, we perform the division on the right side:

x = 5

Boom! We did it! We found the value of x. It's equal to 5. This is the solution to our equation. To recap, we divided both sides of the equation by the coefficient of x (which was 3). This isolates x and gives us its value.

This step highlights the importance of inverse operations in solving equations. Every mathematical operation has an inverse operation that undoes it. Addition and subtraction are inverse operations of each other, and multiplication and division are inverse operations of each other. By understanding these relationships, you can strategically manipulate equations to isolate the variable you're trying to solve for.

Think of it like untangling a knot. You need to identify the last thing that was done to tie the knot and then undo it in reverse order. Similarly, in our equation, x was first multiplied by 3, and then 5 was subtracted. To solve for x, we first undid the subtraction by adding 5, and then we undid the multiplication by dividing by 3. This methodical approach will help you tackle even more complex equations with confidence!

Verification: Making Sure Our Answer is Correct

Now, before we declare victory and move on to the next problem, it's always a good idea to verify our answer. This is like double-checking your work to make sure you didn't make any mistakes. To verify our answer, we simply plug the value we found for x (which is 5) back into the original equation:

3x - 5 = 10

Substitute x = 5:

3 * (5) - 5 = 10

Now, let's simplify the left side of the equation:

15 - 5 = 10

10 = 10

Look at that! The left side of the equation equals the right side of the equation. This means our answer is correct! x = 5 is indeed the solution to the equation 3x - 5 = 10.

Verification is a crucial step in problem-solving because it helps you catch any errors you might have made along the way. It's like proofreading your essay before submitting it. By plugging your answer back into the original equation, you can ensure that your solution satisfies the equation and that you haven't made any arithmetic mistakes.

Furthermore, verification reinforces your understanding of the equation and the solution. It helps you see how the value of x relates to the other terms in the equation and how it makes the equation true. This deeper understanding will make you a more confident and skilled problem-solver in the long run. So, always remember to verify your answers, guys! It's a small step that can make a big difference.

Conclusion

So, there you have it! We successfully solved for x in the equation 3x - 5 = 10. We found that x = 5. We did this by isolating the term with x, then isolating x itself, and finally verifying our answer. Remember the key principles: perform the same operations on both sides of the equation to maintain balance, and use inverse operations to undo operations and isolate the variable.

Understanding these steps will not only help you solve this specific equation but will also equip you with the tools to tackle a wide range of linear equations. The most important thing is to practice! The more you practice, the more comfortable and confident you'll become. And remember, if you get stuck, break the problem down into smaller, more manageable steps.

Keep practicing, and you'll be solving equations like a pro in no time! Good luck, and happy problem-solving!