Solution Set Of 3 - 2x = 7 - 3x A Comprehensive Guide

Hey guys! Ever found yourself scratching your head over a simple equation? Let's break down a classic one today: 3 - 2x = 7 - 3x. We're going to find the solution set and justify the answer step by step. Think of it as untangling a knot – slow, steady, and super satisfying when you get it right! Grab your thinking caps, and let's dive into the world of equations!

Understanding the Problem

Before we jump into solving, let’s make sure we really understand what's being asked. The question gives us the equation 3 - 2x = 7 - 3x and asks for the solution set. What does that even mean? Well, a solution set is simply the set of all values that make the equation true. In this case, we need to find the value (or values) of 'x' that balance both sides of the equation.

The equation itself is a linear equation in one variable. Linear means that the highest power of 'x' is 1 (no x², x³, etc.). One variable means we only have one unknown, which is 'x'. Solving these equations is a fundamental skill in algebra, and it pops up everywhere in math and even in real-world problems. So, mastering this is a big win!

We’re also given some answer choices: A) {x = 4}, B) {x = 1}, C) {x = 2}, and D) {x = 3}. These are potential solutions, but we can't just pick one randomly. We need to solve the equation properly and then see which option matches our solution. This is where the 'justify your answer' part comes in. We're not just guessing; we're showing why our answer is correct. It’s like being a math detective – we need the evidence to back up our claim!

So, the goal is clear: find the value of 'x' that makes the equation true and explain how we found it. Time to roll up our sleeves and get to work!

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation. The key to solving any equation is to isolate the variable – in this case, 'x' – on one side of the equals sign. We want to get 'x' all by itself so we can see what it equals. Think of it like separating the ingredients in a salad; we want the 'x' ingredient on its own plate!

Here's the equation we're starting with:

3 - 2x = 7 - 3x

Step 1: Gather the 'x' terms on one side.

To do this, we want to get all the terms with 'x' on the same side of the equation. A common strategy is to move the term with the smaller coefficient (the number in front of 'x') to the other side. In this case, -3x is smaller than -2x. So, we'll add 3x to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. It’s like a seesaw; if you add weight to one side, you need to add the same weight to the other to keep it level.

Adding 3x to both sides gives us:

3 - 2x + 3x = 7 - 3x + 3x

Simplifying this, we get:

3 + x = 7

Look at that! The '-3x' term has disappeared from the right side, and we've got all our 'x' terms on the left. We’re one step closer to isolating 'x'.

Step 2: Isolate 'x' by moving the constant term.

Now we need to get rid of the '3' on the left side so that 'x' is all alone. Since it's being added to 'x', we'll do the opposite operation: subtract 3 from both sides. Again, balance is key! We need to do the same thing to both sides to maintain the equality.

Subtracting 3 from both sides gives us:

3 + x - 3 = 7 - 3

Simplifying, we get:

x = 4

And there it is! We've done it! We've isolated 'x', and we found that x = 4. It’s like finding the missing piece of a puzzle. We now know the value that makes the equation true.

Verifying the Solution

Before we celebrate too much, it’s always a good idea to double-check our work. We need to verify our solution to make sure it actually works. This is like proofreading an essay or testing a recipe; it’s the final step to ensure we’ve got it right.

To verify our solution, we'll substitute x = 4 back into the original equation:

3 - 2x = 7 - 3x

Replace 'x' with '4':

3 - 2(4) = 7 - 3(4)

Now, we simplify both sides separately:

Left side: 3 - 2(4) = 3 - 8 = -5

Right side: 7 - 3(4) = 7 - 12 = -5

Look at that! Both sides equal -5. This means our solution, x = 4, is correct. The equation balances perfectly when we plug in x = 4. It’s like the two sides of a scale being perfectly level – a satisfying confirmation that we’ve solved it right!

The Solution Set and the Correct Option

Now that we've solved the equation and verified our solution, we know that x = 4 is the value that makes the equation true. This means the solution set is the set containing only the number 4. We write this as {4}.

Let's go back to the answer choices we were given:

A) {x = 4}

B) {x = 1}

C) {x = 2}

D) {x = 3}

Our solution, {4}, matches option A. So, the correct answer is A) {x = 4}. It’s like finding the matching key for a lock – we tried the others, but only this one fits!

Why the Other Options Are Incorrect

It's not enough to know the correct answer; it’s also helpful to understand why the other options are wrong. This can solidify our understanding of the problem and help us avoid similar mistakes in the future. Think of it as learning from our near misses – it makes us stronger!

Let's quickly check why options B, C, and D are incorrect. We can do this by substituting x = 1, x = 2, and x = 3 into the original equation and seeing if the equation holds true.

For x = 1 (Option B):

3 - 2(1) = 7 - 3(1)

3 - 2 = 7 - 3

1 = 4 (This is false)

For x = 2 (Option C):

3 - 2(2) = 7 - 3(2)

3 - 4 = 7 - 6

-1 = 1 (This is false)

For x = 3 (Option D):

3 - 2(3) = 7 - 3(3)

3 - 6 = 7 - 9

-3 = -2 (This is false)

As we can see, none of these values make the equation true. This is why they are not part of the solution set. It’s like trying to fit the wrong puzzle pieces – they just don’t fit!

Conclusion

Alright, guys! We’ve successfully found the solution set for the equation 3 - 2x = 7 - 3x. We started by understanding the problem, then we solved the equation step-by-step, verified our solution, and identified the correct option. We even explored why the other options were incorrect. That’s a pretty thorough job!

Remember, the key to solving equations is to isolate the variable by performing the same operations on both sides of the equation. And always, always verify your solution! It’s like the final brushstroke on a painting – it ensures everything looks perfect.

So, the final answer is A) {x = 4}. You did it! You’re now one equation wiser. Keep practicing, and you’ll become a master equation solver in no time. You've got this!