Solve 12y = 132: The Easy Way
Hey guys! Ever run into a math problem that looks a bit tricky at first glance, but then turns out to be super straightforward? That's exactly what we've got with the equation 12y = 132. We're going to break this down, step-by-step, so you can totally nail it. Our main goal here is to find out what 'y' is equal to. Think of 'y' as a mystery number we need to uncover. The equation tells us that when you take our mystery number 'y' and multiply it by 12, you get 132. So, how do we isolate 'y' and find its value? It's all about using the opposite operation. Since 'y' is being multiplied by 12, we need to divide both sides of the equation by 12. This is a fundamental rule in algebra: whatever you do to one side of the equation, you must do to the other side to keep it balanced. It’s like a seesaw – if you add weight to one side, you have to add the same amount to the other to keep it level. So, let's perform that division: 12y / 12 = 132 / 12. On the left side, the 12s cancel each other out, leaving us with just 'y'. On the right side, we need to calculate 132 divided by 12. If you're not sure, you can do long division, or maybe you recognize that 12 times 10 is 120, and 132 is just 12 more than that. So, 12 times 11 must be 132. Therefore, y = 11. And just like that, our mystery is solved! The value of 'y' is 11. We can quickly check this by plugging 11 back into the original equation: 12 * 11 = 132. It checks out perfectly! So, the answer is y = 11. This method of isolating the variable by using inverse operations is a cornerstone of solving algebraic equations, and it's a skill that will serve you incredibly well as you tackle more complex math challenges. Remember, practice makes perfect, so keep working through these problems, and you'll become a math whiz in no time! Don't be intimidated by letters in equations; they're just placeholders for numbers waiting to be discovered. We're going to look at a few more examples to really cement this concept, so stick around!
Understanding the Equation: 12y = 132
Alright, let's dive a little deeper into what the equation 12y = 132 is actually telling us, guys. This is a basic linear equation, and understanding its structure is key to solving it. The equation is composed of several parts: the coefficient (12), the variable (y), the equals sign (=), and the constant (132). The coefficient is the number multiplying the variable. In this case, it's 12. The variable, represented by 'y', is the unknown value we're trying to find. The equals sign signifies that the expression on the left side (12y) has the exact same value as the expression on the right side (132). Finally, the constant is a fixed number that doesn't change. Our goal in solving this equation is to isolate the variable 'y' on one side of the equals sign. This means we want to get the equation into the form 'y = [some number]'. To do this, we employ the principle of inverse operations. For every mathematical operation, there's an opposite operation that undoes it. Addition's opposite is subtraction, and multiplication's opposite is division. In 12y = 132, the variable 'y' is being multiplied by 12. To undo this multiplication and get 'y' by itself, we need to perform the inverse operation: division. We must divide both sides of the equation by the coefficient, which is 12. This is crucial for maintaining the equality. If we only divided one side, the balance would be broken, and the equation would no longer be true. So, we write it out as: . On the left side, the 12 in the numerator and the 12 in the denominator cancel each other out, leaving us with just 'y'. On the right side, we perform the division: 132 divided by 12. This calculation yields 11. Therefore, y = 11. It’s a pretty elegant process, isn’t it? This fundamental concept of using inverse operations to isolate variables is what allows us to solve a vast array of algebraic problems. Whether you're dealing with simple equations like this or much more complex ones, the underlying logic remains the same. It’s all about balancing the equation and using those opposite operations to uncover the unknown. So, when you see an equation like 12y = 132, don't get flustered. Just remember the goal: get 'y' alone by doing the opposite of what's being done to it. And always, always remember to do it to both sides! Keep practicing these steps, and you'll be solving equations like a pro before you know it. This foundational knowledge is super important for everything that comes next in your math journey, so give yourself a pat on the back for tackling it!
The Mathematical Process: Solving for 'y'
Let's get into the nitty-gritty of the mathematical process for solving 12y = 132. As we've discussed, our mission is to find the value of 'y'. The equation is set up with '12' multiplying 'y' on one side, and '132' on the other. The key principle here is maintaining equality. We need to manipulate the equation so that 'y' stands alone. Since 'y' is currently being multiplied by 12, the inverse operation we need to apply is division. We will divide both sides of the equation by 12. This looks like:
On the left side, the '12' in the numerator and the '12' in the denominator cancel each other out. This is because any number divided by itself equals 1, so we're essentially left with 1*y, which is just 'y'.
Now, we need to perform the division on the right side: 132 divided by 12. Let's think about this. We know that 12 multiplied by 10 is 120. We have 132, which is 12 more than 120 (132 - 120 = 12). So, if 12 * 10 = 120, and we need to add another 12, that means we need to add one more group of 12. Therefore, 12 multiplied by 11 will give us 132.
And there you have it! The solution to the equation 12y = 132 is y = 11. It's a straightforward process once you understand the concept of inverse operations and the importance of balance in an equation. We can verify our answer by substituting 'y = 11' back into the original equation: 12 * 11 = 132. Since 12 * 11 indeed equals 132, our solution is correct. This method is fundamental to algebra and opens the door to solving much more complex problems. Keep practicing these basic steps, and you'll build a solid foundation for advanced math!
Checking Your Answer: Verification Steps
So, you've solved the equation 12y = 132 and you've arrived at y = 11. Awesome job, guys! But how do you know for sure that you're right? This is where the crucial step of verification comes in. It's like double-checking your work before handing in a big test – it gives you confidence in your answer. The process of verification is super simple: you just plug your calculated value of 'y' back into the original equation and see if both sides are equal. Our original equation was 12y = 132. We found that y = 11. So, let's substitute 11 wherever we see 'y' in the original equation:
Now, we perform the multiplication on the left side: 12 multiplied by 11. If you're not instantly sure, you can break it down: 12 * 10 = 120, and 12 * 1 = 12. Add those together: 120 + 12 = 132.
Look at that! The left side of the equation equals the right side. This means our solution, y = 11, is absolutely correct. This verification step is not just for this simple equation; it's a vital practice for all algebraic problem-solving. It helps catch errors and reinforces your understanding of how the equations work. If you had gotten a different answer, say y = 10, and plugged it back in, you'd see: 12 * 10 = 120. Then the equation would read 120 = 132, which is clearly false. That false statement would tell you immediately that you need to go back and re-check your steps. So, always take that extra moment to verify your answers. It's a small step that makes a huge difference in accuracy and confidence. Keep this verification habit going, and you'll be acing your math assignments!
Comparing with Options
Now that we've diligently solved the equation 12y = 132 and confirmed that y = 11, let's take a look at the multiple-choice options provided. This is where you match your hard-earned answer to the correct selection. The options are:
A. $y=\frac{11}{12}$ B. $y=\frac{1}{11}$ C. $y=11$ D. $y=144$
We found our solution to be y = 11. Let's compare this directly with each option.
- Option A suggests $y=\frac{11}{12}$. This is a fraction, and it's not equal to 11. So, this option is incorrect.
- Option B suggests $y=\frac{1}{11}$. Again, this is a fraction and quite different from our calculated value. This option is also incorrect.
- Option C suggests y = 11. This exactly matches the value we calculated through our step-by-step solving process and verification. This is our correct answer!
- Option D suggests $y=144$. This number is significantly larger than our solution. If we were to check this: 12 * 144 would be a very large number, definitely not 132. So, this option is incorrect.
Therefore, the correct answer is Option C, y = 11. It's always satisfying when your calculated answer perfectly aligns with one of the given choices. This confirms that your understanding and application of the solving steps were spot on. Keep up the great work, guys, and remember to always trust your calculations after performing verification!