Solving For X: 1 - 3x = 3x + 1 Equation Explained

by SLV Team 50 views
Solving for x: 1 - 3x = 3x + 1 Equation Explained

Hey guys! Let's dive into solving a classic algebraic equation today. We're going to break down the steps to solve for x in the equation 1 - 3x = 3x + 1. This type of problem is fundamental in algebra, and understanding how to solve it will help you tackle more complex equations down the road. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into the solution, it's essential to understand what the equation is telling us. The equation 1 - 3x = 3x + 1 is a linear equation, meaning that the highest power of x is 1. Our goal is to isolate x on one side of the equation to find its value. Think of it like a balancing act – we need to perform operations on both sides of the equation to maintain the balance while getting x by itself.

When we're faced with an equation like this, the main idea is to consolidate like terms. What does that mean? Well, we want all the terms with x on one side and all the constant terms (the numbers) on the other side. This makes the equation cleaner and easier to solve. There are several steps involved, and each one is crucial for arriving at the correct answer. We'll go through each step in detail, so don't worry if it seems a bit confusing at first. The key is to follow along and understand the logic behind each move. Algebra is like a puzzle, and each step is a piece that fits into the bigger picture.

Key Concepts to Remember:

  • Linear Equation: An equation where the highest power of the variable is 1.
  • Isolate the Variable: The goal is to get the variable (in this case, x) by itself on one side of the equation.
  • Like Terms: Terms that have the same variable raised to the same power (e.g., 3x and -3x) or are constants (e.g., 1 and -1).
  • Maintaining Balance: Whatever operation you perform on one side of the equation, you must perform on the other side to keep the equation balanced.

Step-by-Step Solution

Now, let's get to the fun part – solving the equation! We'll take it one step at a time, explaining the reasoning behind each step. Trust me, once you get the hang of it, you'll be solving equations like a pro.

Step 1: Moving the x Terms

The first thing we want to do is get all the x terms on one side of the equation. Looking at our equation, 1 - 3x = 3x + 1, we have -3x on the left side and 3x on the right side. A common strategy is to move the term with the smaller coefficient (the number in front of x) to the side with the larger coefficient. In this case, we'll add 3x to both sides of the equation to eliminate the -3x term on the left. Here’s how it looks:

1 - 3x + 3x = 3x + 1 + 3x

By adding 3x to both sides, we maintain the balance of the equation. On the left side, -3x and +3x cancel each other out, leaving us with just 1. On the right side, 3x + 3x combines to give us 6x. This simplifies our equation to:

1 = 6x + 1

See? We're already making progress! The equation is looking simpler, and we're one step closer to isolating x. This step is crucial because it consolidates our x terms into one place, which makes the next steps much easier to manage. Think of it as gathering all the ingredients you need before you start cooking – it just makes the process smoother.

Step 2: Moving the Constant Terms

Next up, we need to move all the constant terms (the numbers without x) to the other side of the equation. In our simplified equation, 1 = 6x + 1, we have a +1 on both sides. To get rid of the +1 on the right side, we'll subtract 1 from both sides. This keeps the equation balanced and helps us isolate the term with x. Here’s the magic:

1 - 1 = 6x + 1 - 1

Subtracting 1 from both sides does the trick. On the left side, 1 - 1 equals 0. On the right side, +1 and -1 cancel each other out, leaving us with just 6x. Our equation now looks like this:

0 = 6x

Wow, it's getting even simpler! We’ve managed to get all the constant terms on one side, and now we have a very straightforward equation to solve. This step is all about decluttering the equation and making it easier to see the next move. It’s like clearing your desk before you start working – a clean space helps you focus better.

Step 3: Isolating x

Now comes the final step – isolating x. We have the equation 0 = 6x. This means 6 times x equals 0. To find the value of x, we need to undo the multiplication by 6. We do this by dividing both sides of the equation by 6. Are you ready for it?

0 / 6 = 6x / 6

Dividing both sides by 6 is the key to unlocking x. On the left side, 0 divided by any non-zero number is 0. On the right side, 6x divided by 6 simplifies to just x. So, we end up with:

0 = x

Or, if we prefer to write it the other way around:

x = 0

And there you have it! We've solved for x. The value of x that satisfies the equation 1 - 3x = 3x + 1 is 0. Congratulations, you did it! This step is the culmination of all our efforts, and it's so satisfying to see that single value of x that makes the equation true.

Checking the Solution

It's always a good idea to check your solution to make sure it's correct. To do this, we substitute the value we found for x (which is 0) back into the original equation, 1 - 3x = 3x + 1. If both sides of the equation are equal after the substitution, then our solution is correct. Let’s plug it in:

1 - 3(0) = 3(0) + 1

Now, let's simplify each side:

On the left side:

1 - 3(0) = 1 - 0 = 1

On the right side:

3(0) + 1 = 0 + 1 = 1

So, we have:

1 = 1

Both sides of the equation are equal! This confirms that our solution, x = 0, is indeed correct. Checking your solution is like proofreading your work – it’s an extra step that ensures you haven’t made any mistakes along the way. It gives you confidence in your answer and helps reinforce your understanding of the problem.

Alternative Methods

While we've walked through one way to solve the equation, it's worth noting that there are often multiple paths to the same answer in algebra. Let's briefly touch on an alternative approach to solving 1 - 3x = 3x + 1. Instead of moving the -3x term first, we could have chosen to move the constant term first. Here’s how that would look:

Alternative Method: Moving Constants First

  1. Subtract 1 from both sides:

    1 - 3x - 1 = 3x + 1 - 1

    This simplifies to:

    -3x = 3x

  2. Subtract 3x from both sides:

    -3x - 3x = 3x - 3x

    This simplifies to:

    -6x = 0

  3. Divide both sides by -6:

    -6x / -6 = 0 / -6

    This gives us:

    x = 0

As you can see, we arrived at the same solution, x = 0, but by taking a slightly different route. This illustrates a key point in algebra: there's often more than one way to solve a problem. The important thing is to choose a method that makes sense to you and follow the rules of algebra to maintain the balance of the equation.

Real-World Applications

You might be wondering,