Solving 10 - (2/3)(3x - 12) = 2: A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks like a total headache? Well, today we’re diving deep into one that might seem tricky at first glance: 10 - (2/3)(3x - 12) = 2. Don't worry; we're going to break it down step-by-step, so by the end of this guide, you'll be solving these like a pro! So grab your pencils, notebooks, and let’s get started! We will be focusing on how to solve this specific equation. Equations like this aren't as scary as they look. Think of it as a puzzle. You just need to untangle the pieces carefully. The key is to take it one step at a time and not rush the process. Trust me; with a little patience and understanding, you can conquer any algebraic challenge. Remember, mathematics isn't about memorizing formulas; it's about understanding the logic behind them. Once you get that, you'll find that solving equations becomes more intuitive and, dare I say, even fun! So, let's get into it and show you how to crack this particular nut. We're going to take you through each stage, explaining why we do what we do. This way, you're not just copying steps; you're learning the principles. This approach will help you tackle similar problems in the future with confidence. So, stick with us, and let's transform this equation from a daunting task into a simple, satisfying victory. Ready? Let's roll!

Understanding the Equation

Before we start punching in numbers, let's take a moment to understand what this equation, 10 - (2/3)(3x - 12) = 2, is all about. At its heart, an equation is a statement that two things are equal. Our job is to find the value of 'x' that makes this statement true.

Breaking it down:

• Constants: We have constants like 10 and 2, which are just regular numbers.
• Variable: We have 'x', which is the unknown value we're trying to find.
• Operations: We have operations like subtraction, multiplication, and division that connect these terms.

The term (2/3)(3x - 12) is particularly important. It means we're taking two-thirds of the quantity (3x - 12). The parentheses indicate that we need to deal with what's inside them before we can multiply by (2/3). This is where the order of operations comes into play, something we'll use throughout the solution.

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), tells us the sequence in which we should perform operations. Ignoring this order can lead to incorrect answers, so keep it in mind. Also, remember that whatever we do on one side of the equation, we must also do on the other side to maintain the balance. This principle is fundamental to solving equations because it ensures that the equality remains valid. With this understanding, we're well-equipped to start simplifying and solving for 'x'.

Step 1: Distribute the (2/3)

The first thing we need to do is get rid of those parentheses in the equation 10 - (2/3)(3x - 12) = 2. To do that, we'll distribute the (2/3) to both terms inside the parentheses.

(2/3) * (3x) = 2x (2/3) * (-12) = -8

So, our equation now looks like this: 10 - (2x - 8) = 2. Be super careful with that minus sign in front of the parentheses! It's like a sneaky ninja that can change the signs of everything inside. We're essentially subtracting the entire quantity (2x - 8), so we need to distribute the negative sign as well.

Watch out for the minus! Distributing the negative sign is a common place to make mistakes. Remember, subtracting a negative is the same as adding, so -(2x - 8) becomes -2x + 8.

Our equation now transforms to: 10 - 2x + 8 = 2. By distributing the (2/3) and carefully managing the negative sign, we've cleared the first hurdle and simplified our equation, making it easier to work with.

Step 2: Combine Like Terms

Alright, now that we've distributed and gotten rid of those pesky parentheses, it's time to combine the like terms in our equation: 10 - 2x + 8 = 2.

Like terms are terms that have the same variable raised to the same power. In this case, 10 and 8 are like terms because they're both constants (numbers without any variables attached).

Combining 10 and 8:

10 + 8 = 18

So, we can rewrite our equation as:

18 - 2x = 2

Combining like terms helps simplify the equation by reducing the number of terms we have to deal with. It makes the equation cleaner and easier to manipulate. In this step, we've essentially tidied up our equation, making it more manageable for the next steps. It's like organizing your workspace before tackling a bigger task – it just makes everything smoother. By adding the constants together, we've taken another step toward isolating 'x' and solving for its value. This process of simplification is crucial in algebra because it breaks down complex equations into more understandable and solvable forms.

Step 3: Isolate the Variable Term

Now we need to isolate the term with 'x' in the equation 18 - 2x = 2. This means we want to get the -2x term all by itself on one side of the equation. To do that, we need to get rid of the 18 that's hanging out with it.

Subtracting 18 from both sides:

To remove the 18, we'll subtract 18 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.

18 - 2x - 18 = 2 - 18

This simplifies to:

-2x = -16

By subtracting 18 from both sides, we've successfully isolated the term with 'x'. Now we have a much simpler equation to work with. Isolating the variable term is a critical step in solving equations because it brings us closer to finding the value of the variable. It's like peeling away the layers of an onion – each step gets us closer to the core. Now that we have -2x = -16, we're just one step away from finding the value of 'x'.

Step 4: Solve for x

We're almost there! We've got -2x = -16, and our goal is to find what 'x' equals. Right now, 'x' is being multiplied by -2, so to isolate 'x', we need to do the opposite operation: divide both sides of the equation by -2.

Dividing both sides by -2:

(-2x) / -2 = (-16) / -2

This simplifies to:

x = 8

And that's it! We've solved for 'x'. The value of 'x' that makes the original equation true is 8. To verify our answer, we can substitute x = 8 back into the original equation to make sure it holds.

Step 5: Check Your Answer

To make sure we didn't make any mistakes along the way, let's plug our solution, x = 8, back into the original equation: 10 - (2/3)(3x - 12) = 2.

Substituting x = 8:

10 - (2/3)(3(8) - 12) = 2

Now, let's simplify step-by-step:

10 - (2/3)(24 - 12) = 2 10 - (2/3)(12) = 2 10 - 8 = 2 2 = 2

Our solution checks out! Both sides of the equation are equal, which means x = 8 is indeed the correct answer.

Why is checking important? Checking your answer is a crucial step in solving equations. It helps you catch any errors you might have made during the solving process. It's like proofreading an essay before submitting it – you want to make sure everything is correct and makes sense. By verifying your solution, you can have confidence that you've solved the equation correctly. This also reinforces your understanding of the steps involved and helps solidify your problem-solving skills.

Conclusion

So, there you have it! We've successfully solved the equation 10 - (2/3)(3x - 12) = 2, and found that x = 8. Remember, the key to solving equations is to take it step-by-step, simplify as much as possible, and always check your answer. With practice, you'll become more confident and proficient in solving algebraic equations. Keep practicing, and don't be afraid to tackle challenging problems. You got this! Now you are ready to solve more equations! You are getting better at math everyday. Keep it up and you will be an expert in no time! Always remember to check your answers and don't be afraid to ask for help.