Solve 3x - 12 = 97: Step-by-Step Guide

by Andrew McMorgan 39 views

Hey math whizzes and curious minds of Plastik Magazine! Ever stare at an equation and feel like you're deciphering ancient hieroglyphs? Yeah, we've all been there. Today, we're diving deep into a classic: solving the equation 3x - 12 = 97. Forget the panic, guys, because by the end of this, you'll be an algebra ninja, ready to tackle any linear equation that comes your way. We'll break it down step-by-step, explaining the 'why' behind each move, so you truly understand the magic of isolating that elusive 'x'. So grab your calculators (or just your brains!), and let's get this solved!

Understanding the Goal: Isolating 'x'

The fundamental goal when you're faced with an equation like 3x - 12 = 97 is to figure out the value of the unknown variable, which is 'x' in this case. Think of it like a puzzle where 'x' is the missing piece. To find 'x', we need to get it all by itself on one side of the equals sign. This process is called isolating the variable. We do this by performing inverse operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? When solving equations, we often work in the reverse order of operations, starting with addition and subtraction, and then moving to multiplication and division. This ensures that we maintain the balance of the equation – whatever we do to one side, we must do to the other. It's like a perfectly balanced scale; if you add weight to one side, you have to add the same weight to the other to keep it level. So, for 3x - 12 = 97, our mission is to peel away the '-12' and the '3' that are attached to 'x', until 'x' is standing alone, proud and solved. We'll use addition, subtraction, multiplication, and division strategically to achieve this, always keeping that equation balanced. The power lies in the inverse operations: addition undoes subtraction, subtraction undoes addition, division undoes multiplication, and multiplication undoes division. Mastering this concept is key to unlocking the secrets of algebraic equations, not just this one, but countless others you'll encounter.

Step 1: Undoing Subtraction

Alright, let's get down to business with 3x - 12 = 97. Our first target is that '-12'. Remember, we want 'x' all by its lonesome. The opposite (or inverse operation) of subtracting 12 is adding 12. So, to get rid of the '-12' on the left side of the equation, we're going to add 12 to both sides. This is crucial for keeping the equation balanced. Think of it as giving both sides a little boost of +12. On the left side, '-12 + 12' cancels each other out, leaving us with just '3x'. On the right side, we have '97 + 12'. Let's do that math: 97 plus 10 is 107, and then add the remaining 2, which brings us to 109. So, after this first step, our equation transforms from 3x - 12 = 97 into 3x = 109. See? We're already one step closer to finding 'x'. This might seem simple, but this principle of using inverse operations to eliminate terms is the bedrock of solving all sorts of algebraic problems. It's the fundamental move that allows us to simplify complex expressions and eventually pinpoint the value of our unknown. We've successfully neutralized the constant term on the side with our variable, setting the stage for the next move. This is where the 'peeling the onion' analogy really comes into play – we're taking off layers to get to the core.

Step 2: Undoing Multiplication

Now we're looking at 3x = 109. We've successfully eliminated the '-12', but 'x' is still not alone. It's being multiplied by 3. Remember, when a number is right next to a variable like '3x', it means multiplication (3 * x). To undo multiplication, we use its inverse operation: division. So, we need to divide both sides of the equation by 3. This is our balancing act again! On the left side, dividing '3x' by 3 leaves us with just 'x' (because 3 divided by 3 is 1, and 1x is just x). On the right side, we need to divide 109 by 3. Now, let's see if 109 divides evenly by 3. We can use the divisibility rule for 3: add the digits of the number. 1 + 0 + 9 = 10. Since 10 is not divisible by 3, 109 is not divisible by 3. This means our answer will be a fraction or a decimal. So, x = 109/3. If we want to express this as a mixed number, we can divide 109 by 3. 3 goes into 10 three times (33=9), with a remainder of 1. Bring down the 9, making it 19. 3 goes into 19 six times (36=18), with a remainder of 1. So, 109/3 is equal to 36 and 1/3, or approximately 36.33. The exact answer is 109/3. This step is just as critical as the first. We've isolated 'x' by undoing the multiplication. The fact that it's not a whole number isn't a problem; it just means our solution is a rational number. Always be prepared for fractions or decimals in algebra – they are perfectly valid solutions! The key takeaway here is that division is the direct counter to multiplication, allowing us to isolate our variable when it's being acted upon by multiplication.

Checking Our Answer

So, we found that x = 109/3. But are we sure? In the world of math, especially algebra, checking your work is super important. It's your chance to be your own math detective and make sure you haven't made any sneaky errors. To check our solution, we substitute our value of 'x' back into the original equation: 3x - 12 = 97. So, we'll replace 'x' with '109/3'. This gives us: 3 * (109/3) - 12 = 97. Now, let's simplify the left side. The '3' in front of the parenthesis multiplies the '109/3'. Since we have a 3 in the numerator and a 3 in the denominator, they cancel each other out, leaving us with just 109. So, the equation becomes 109 - 12 = 97. Now, let's do the subtraction: 109 minus 12. That's 97. So, we have 97 = 97. Ta-da! The left side equals the right side. This confirms that our solution, x = 109/3, is absolutely correct. This checking process is a lifesaver, guys. It builds confidence in your answers and helps you catch mistakes before they become bigger problems. Never skip this step when you can! It transforms uncertainty into certainty and solidifies your understanding of the equation's solution. It's the final stamp of approval that your algebraic journey was a success for this particular problem.

Multiple Choice Options and Final Answer

We've worked through the equation 3x - 12 = 97 and found our solution to be x = 109/3. Now, let's look at the multiple-choice options provided: A. 5, B. 7, C. 15, D. -5. Hmm, our calculated answer, 109/3 (which is approximately 36.33), doesn't directly match any of these whole numbers. This is a common scenario in standardized tests or problem sets where perhaps the original problem was slightly different, or the options are designed to catch common mistakes. Let's re-evaluate our steps to ensure accuracy and consider if there might be a typo in the question or options given. Our calculations were:

  1. Add 12 to both sides: 3x = 97 + 12 = 109
  2. Divide both sides by 3: x = 109/3

These steps are mathematically sound for the equation 3x - 12 = 97. If we were forced to choose from the options, we'd have to suspect an error in the question's formulation or the provided choices, as none of them yield 97 when plugged back into the original equation.

Let's test the options to see what value they would produce:

  • If x = 5: 3(5) - 12 = 15 - 12 = 3 (Not 97)
  • If x = 7: 3(7) - 12 = 21 - 12 = 9 (Not 97)
  • If x = 15: 3(15) - 12 = 45 - 12 = 33 (Not 97)
  • If x = -5: 3(-5) - 12 = -15 - 12 = -27 (Not 97)

It appears there might be a discrepancy between the equation provided and the multiple-choice answers. However, if we assume there was a typo and the equation was meant to lead to one of these answers, let's consider what equation would result in one of the options. For example, if the answer was meant to be 15 (Option C), the equation would need to be something like 3x - 12 = 33. Since our thorough step-by-step calculation for 3x - 12 = 97 yielded x = 109/3, and this does not match any of the given options, we must conclude that based on the exact equation provided, none of the options are correct. In a real test scenario, you might flag this question or re-read it very carefully for any missed details. For the purpose of this guide, our derived solution is x = 109/3.

Conclusion: Mastering Linear Equations

So there you have it, folks! We've successfully navigated the process of solving the linear equation 3x - 12 = 97. We learned the critical importance of isolating the variable by using inverse operations to maintain the balance of the equation. From undoing subtraction by adding 12 to both sides, to undoing multiplication by dividing both sides by 3, we arrived at the solution x = 109/3. We even took the crucial step of checking our answer by substituting it back into the original equation, confirming its accuracy when we got 97 = 97. While the provided multiple-choice options didn't align with our calculated result, our methodical approach ensures we identified the correct solution for the equation as written. This process isn't just about solving one problem; it's about building a foundational skill in algebra. Whether you're dealing with simple linear equations or more complex problems down the line, the principles of inverse operations and maintaining balance are your best friends. Keep practicing, keep questioning, and don't be afraid of fractions or decimals – they're just numbers, too! You've got this, Plastik Magazine readers. Keep those brains buzzing!