Solving Equations: Find X In 1.2x = 5.16
Alright, let's dive into solving this equation! When we're faced with something like 1.2x = 5.16, our main goal is to isolate x on one side of the equation. This means we want to get x all by itself so we can clearly see what its value is. It's like finding the missing piece of a puzzle, and in this case, x is that missing piece. To kick things off, remember that 1.2x actually means 1.2 multiplied by x. So, to undo this multiplication, we need to perform the opposite operation, which is division. We're going to divide both sides of the equation by 1.2. This keeps the equation balanced, which is super important. Think of it like a seesaw – whatever you do to one side, you have to do to the other to keep it level. If we only divided one side, the equation would be unbalanced, and we wouldn't get the correct value for x. By dividing both sides, we ensure that the equation remains true and that we're on the right track to finding our solution.
Step-by-Step Solution
So, let's get into the nitty-gritty of how to solve for x in the equation 1.2x = 5.16. First things first, write down the equation so we can see exactly what we're working with: 1.2x = 5.16. Now, remember that our goal is to isolate x. This means we want to get x by itself on one side of the equation. To do this, we need to undo the multiplication that's happening between 1.2 and x. The opposite of multiplication is division, so we're going to divide both sides of the equation by 1.2. When we divide both sides by 1.2, we get: 1.2x / 1.2 = 5.16 / 1.2. On the left side of the equation, the 1.2 in the numerator and the 1.2 in the denominator cancel each other out, leaving us with just x. This is exactly what we wanted! Now we have: x = 5.16 / 1.2. All that's left to do is perform the division on the right side of the equation. When we divide 5.16 by 1.2, we get 4.3. So, our final answer is: x = 4.3. This means that the value of x that makes the equation 1.2x = 5.16 true is 4.3. We've successfully solved for x!
Performing the Division
Now, let's break down the division of 5.16 by 1.2 to make sure we understand exactly how we arrived at the answer of 4.3. When we're dividing decimals, it can sometimes be a bit tricky, but there's a simple trick to make it easier. The first thing we want to do is get rid of the decimal in the divisor, which in this case is 1.2. To do this, we can multiply both the divisor and the dividend by 10. This moves the decimal point one place to the right in both numbers. So, 1.2 becomes 12, and 5.16 becomes 51.6. Now we have a much simpler division problem: 51.6 / 12. Next, we perform the division as we normally would. We ask ourselves, how many times does 12 go into 51? The answer is 4 times, because 4 * 12 = 48. So we write down the 4 above the 1 in 51.6. Then we subtract 48 from 51, which gives us 3. Now we bring down the 6 from 51.6 to get 36. We ask ourselves, how many times does 12 go into 36? The answer is 3 times, because 3 * 12 = 36. So we write down the 3 after the decimal point above the 6 in 51.6. Finally, we subtract 36 from 36, which gives us 0. This means we've completed the division, and our answer is 4.3. Therefore, 5.16 / 1.2 = 4.3, which confirms our earlier result. Understanding how to perform this division is crucial for solving similar equations in the future.
Verification
To make absolutely sure that our answer is correct, it's always a good idea to verify it. This means we're going to plug our solution, x = 4.3, back into the original equation, 1.2x = 5.16, and see if it holds true. When we substitute 4.3 for x in the equation, we get: 1.2 * 4.3 = 5.16. Now we need to perform the multiplication on the left side of the equation. When we multiply 1.2 by 4.3, we get 5.16. So the equation becomes: 5.16 = 5.16. This is a true statement, which means that our solution, x = 4.3, is indeed correct. Verifying our answer is a crucial step in the problem-solving process. It helps us catch any mistakes we might have made along the way and ensures that we're confident in our final answer. By plugging our solution back into the original equation and confirming that it holds true, we can be certain that we've solved the equation correctly.
Alternative Methods
While dividing both sides of the equation by 1.2 is the most straightforward method for solving 1.2x = 5.16, there are a couple of alternative approaches we could take. One option is to convert the decimal numbers into fractions. 1.2 can be written as 12/10, and 5.16 can be written as 516/100. So our equation becomes: (12/10)x = 516/100. To solve for x, we can multiply both sides of the equation by the reciprocal of 12/10, which is 10/12. This gives us: x = (516/100) * (10/12). We can simplify this expression by canceling out common factors. The 10 in the numerator and the 100 in the denominator cancel out, leaving us with 10 in the denominator. So we have: x = 516 / (10 * 12). 10 * 12 = 120, so we have: x = 516 / 120. Now we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 12. 516 / 12 = 43, and 120 / 12 = 10. So we have: x = 43/10, which is equal to 4.3. This method gives us the same answer as before, but it involves working with fractions instead of decimals. Another alternative approach is to use logarithms, but that's generally not necessary for such a simple equation. Dividing both sides by the coefficient of x is usually the most efficient and straightforward method.
Conclusion
In summary, to solve the equation 1.2x = 5.16 for x, the most efficient method is to divide both sides of the equation by 1.2. This isolates x on one side of the equation and gives us the solution x = 4.3. We can verify this solution by plugging it back into the original equation and confirming that it holds true. While there are alternative methods for solving this equation, such as converting decimals to fractions, dividing both sides by the coefficient of x is generally the most straightforward and efficient approach. Remember to always double-check your work and verify your solution to ensure accuracy. Solving equations like this is a fundamental skill in mathematics, and mastering it will help you tackle more complex problems in the future. Keep practicing, and you'll become a pro at solving for x in no time! So, next time you see an equation like 1.2x = 5.16, you'll know exactly what to do. Just remember to isolate x by performing the opposite operation, and don't forget to verify your answer. Happy problem-solving, guys!