Solving For X: (1/3)x - (2/3) = -18

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Hey guys! Let's break down this math problem step-by-step. We're trying to find the value of x in the equation (1/3)x - (2/3) = -18. Don't worry, it's not as scary as it looks! We will guide you through the process of solving this equation, ensuring you understand each step. This involves isolating x on one side of the equation by performing algebraic operations. So, let's dive in and get started!

Understanding the Equation

First, let's get a good grasp of what the equation is telling us. We have a fraction multiplied by x, then we're subtracting another fraction, and the whole thing equals -18. Our mission is to isolate x, which means getting it all by itself on one side of the equals sign. To do this, we'll need to undo the operations that are being performed on x, one step at a time. This involves using inverse operations to maintain the balance of the equation. Remember, what we do to one side, we must do to the other!

Isolating the Term with x

The first thing we want to do is get rid of that pesky -2/3. How do we do that? By adding its opposite, which is +2/3, to both sides of the equation. Think of it like this: if we add 2/3 to -2/3, they cancel each other out, leaving us with zero. But remember, we have to add 2/3 to both sides to keep the equation balanced. So, here's how it looks:

(1/3)x - (2/3) + (2/3) = -18 + (2/3)

On the left side, the -2/3 and +2/3 cancel out, leaving us with just (1/3)x. On the right side, we need to add -18 and 2/3. To do that, it's helpful to convert -18 into a fraction with a denominator of 3. We know that 18 is the same as 18/1, so we can multiply both the numerator and denominator by 3 to get 54/3. So, -18 is the same as -54/3. Now we can add:

-54/3 + 2/3 = -52/3

So, our equation now looks like this:

(1/3)x = -52/3

Solving for x

We're almost there! Now we have (1/3)x equals -52/3. But we want to know what x equals, not one-third of x. So, we need to get rid of that 1/3 that's multiplying x. How do we do that? By multiplying both sides of the equation by the reciprocal of 1/3, which is 3/1, or just 3. This is because multiplying a fraction by its reciprocal gives us 1.

So, let's multiply both sides by 3:

3 * (1/3)x = 3 * (-52/3)

On the left side, the 3 and 1/3 cancel each other out, leaving us with just x. On the right side, we're multiplying 3 by -52/3. We can think of 3 as 3/1, so we're multiplying (3/1) * (-52/3). When multiplying fractions, we multiply the numerators and multiply the denominators:

(3 * -52) / (1 * 3) = -156 / 3

Now we need to simplify -156/3. We can do this by dividing -156 by 3, which gives us -52. So, the equation becomes:

x = -52

The Answer

And there you have it! We've solved for x. The value of x in the equation (1/3)x - (2/3) = -18 is -52. So, the correct answer is B. -52.

Checking Our Work

It's always a good idea to check our answer to make sure we didn't make any mistakes along the way. To do that, we can plug -52 back into the original equation and see if it holds true. So, let's do it:

(1/3) * (-52) - (2/3) = -18

First, we multiply (1/3) by -52. We can think of -52 as -52/1, so we're multiplying (1/3) * (-52/1). Multiply the numerators and multiply the denominators:

(1 * -52) / (3 * 1) = -52/3

So, the equation now looks like this:

-52/3 - 2/3 = -18

Now we're subtracting 2/3 from -52/3. Since they have the same denominator, we can simply subtract the numerators:

(-52 - 2) / 3 = -54/3

Now we simplify -54/3 by dividing -54 by 3, which gives us -18. So, the equation becomes:

-18 = -18

It checks out! This confirms that our answer of x = -52 is correct.

Key Takeaways for Solving Equations

Solving algebraic equations can seem daunting at first, but with a few key principles in mind, you can tackle them with confidence. Here are some takeaways to remember:

  1. Isolate the Variable: The primary goal in solving any equation is to isolate the variable you're solving for. This means getting the variable by itself on one side of the equation.
  2. Use Inverse Operations: To isolate the variable, use inverse operations. Addition and subtraction are inverse operations, as are multiplication and division. Apply these operations to both sides of the equation to maintain balance.
  3. Maintain Balance: Whatever operation you perform on one side of the equation, you must perform on the other side. This ensures the equation remains balanced and the equality holds true.
  4. Simplify as You Go: Simplify the equation at each step by combining like terms and reducing fractions. This makes the equation easier to work with and reduces the chance of errors.
  5. Check Your Answer: Always check your answer by substituting it back into the original equation. This helps verify that your solution is correct.

By following these principles, you can approach any algebraic equation with a clear strategy and increase your chances of finding the correct solution. Remember, practice makes perfect, so the more equations you solve, the more comfortable and confident you'll become!

Practice Problems

Want to test your skills further? Try solving these similar equations:

  1. (1/4)x + (3/4) = -10
  2. (2/5)x - (1/5) = -7
  3. (1/2)x + (1/4) = -9

Solving these practice problems will help reinforce your understanding and improve your equation-solving abilities. Don't be afraid to make mistakes; they're a natural part of the learning process. The key is to learn from your mistakes and keep practicing!

Conclusion

Solving for x in the equation (1/3)x - (2/3) = -18 involved a few simple steps: adding 2/3 to both sides to isolate the term with x, and then multiplying both sides by 3 to solve for x. We found that x equals -52. Remember, the key to solving equations is to isolate the variable by using inverse operations and keeping the equation balanced. With a little practice, you'll be solving equations like a pro! Keep practicing, and you'll master these skills in no time. You've got this!