Solving The Equation (12y-1)/2 = (9y+8)/5 A Step-by-Step Guide

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Hey guys! Let's dive into this math problem where we need to find the value of y that makes the equation (12y - 1)/2 = (9y + 8)/5 true. Don't worry, we'll break it down step by step so it's super easy to follow. We'll explore different methods to tackle this equation and make sure you understand every part of the process. Math can be fun, especially when we solve these kinds of puzzles together!

Understanding the Equation

First, let's get a good grasp of the equation we're dealing with: (12y - 1)/2 = (9y + 8)/5. This is a linear equation, which means the highest power of y is 1. To solve for y, we need to isolate it on one side of the equation. This involves getting rid of the fractions and then rearranging the terms to get y by itself. We'll use some basic algebraic principles to make this happen, like cross-multiplication and combining like terms. Remember, the goal is to find the value of y that makes both sides of the equation equal.

Step-by-Step Solution

Step 1: Eliminate the Fractions

The first thing we want to do is get rid of those pesky fractions. To do this, we can use a method called cross-multiplication. This means we'll multiply the numerator of the left side by the denominator of the right side, and vice versa. So, we have:

5 * (12y - 1) = 2 * (9y + 8)

This step helps us transform the equation into a more manageable form without fractions.

Step 2: Distribute

Next, we need to distribute the numbers outside the parentheses to the terms inside. This means we multiply 5 by both 12y and -1 on the left side, and we multiply 2 by both 9y and 8 on the right side:

(5 * 12y) - (5 * 1) = (2 * 9y) + (2 * 8)

This simplifies to:

60y - 5 = 18y + 16

Now we have a linear equation without fractions, which is much easier to work with.

Step 3: Combine Like Terms

Our next goal is to get all the y terms on one side of the equation and all the constant terms on the other side. To do this, we'll subtract 18y from both sides and add 5 to both sides:

60y - 18y = 16 + 5

This gives us:

42y = 21

We've successfully combined the like terms and now we're one step closer to finding y.

Step 4: Isolate y

Finally, to isolate y, we need to divide both sides of the equation by 42:

y = 21 / 42

This simplifies to:

y = 1/2

So, the value of y that makes the equation true is 1/2. We've found our answer!

Checking the Solution

To make sure we got the correct answer, it's always a good idea to plug the value of y back into the original equation. Let's substitute y = 1/2 into (12y - 1)/2 = (9y + 8)/5:

Left side:

(12 * (1/2) - 1) / 2 = (6 - 1) / 2 = 5 / 2

Right side:

(9 * (1/2) + 8) / 5 = (4.5 + 8) / 5 = 12.5 / 5 = 2.5

Converting 2.5 to a fraction gives us 5/2. Since both sides of the equation are equal when y = 1/2, our solution is correct. Yay!

Alternative Methods to Solve the Equation

Method 1: Multiplying by the Least Common Multiple (LCM)

Another way to eliminate fractions is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 2 and 5, so the LCM is 10. Let's try this method:

Multiply both sides by 10:

10 * [(12y - 1) / 2] = 10 * [(9y + 8) / 5]

Simplify:

5 * (12y - 1) = 2 * (9y + 8)

Notice that this is the same equation we got after cross-multiplication. From here, we would follow the same steps as before to solve for y.

Method 2: Clearing One Fraction at a Time

Sometimes, you might prefer to clear one fraction at a time. To do this, you can multiply both sides of the equation by the denominator of one fraction and then repeat the process for the other fraction. For example:

Start with the original equation:

(12y - 1) / 2 = (9y + 8) / 5

Multiply both sides by 2:

12y - 1 = (2 * (9y + 8)) / 5

Now, multiply both sides by 5:

5 * (12y - 1) = 2 * (9y + 8)

Again, we end up with the same equation as before. This method is a bit more step-by-step and can be helpful if you find it easier to manage one fraction at a time.

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes that students often make. Let's go over these so you can avoid them:

  1. Forgetting to Distribute: One of the most common errors is not distributing properly. Make sure you multiply the number outside the parentheses by every term inside. For example, when we have 5 * (12y - 1), you need to multiply 5 by both 12y and -1. Forgetting to do this can lead to an incorrect answer.

  2. Incorrectly Combining Like Terms: Be careful when combining like terms. Remember to only combine terms that have the same variable and exponent. For instance, you can combine 60y and -18y, but you can't combine 60y and -5 because they are not like terms. Also, pay attention to the signs (+ and -) when combining terms.

  3. Dividing Incorrectly: When isolating y, make sure you divide both sides of the equation by the correct number. For example, if you have 42y = 21, you need to divide both sides by 42. Sometimes students mistakenly divide only one side or divide by the wrong number, which leads to an incorrect solution.

  4. Not Checking the Solution: Always, always, always check your solution by plugging it back into the original equation. This is the best way to catch any mistakes you might have made. If the left side of the equation equals the right side when you substitute your value of y, you know your answer is correct. If not, you need to go back and find your mistake.

  5. Sign Errors: Sign errors are super common in algebra. Be extra careful with negative signs. For instance, when you move a term from one side of the equation to the other, remember to change its sign. So, if you have 60y - 5 = 18y + 16, when you subtract 18y from both sides, it becomes -18y on the left side.

Real-World Applications

Solving linear equations like this isn't just a math exercise; it has tons of real-world applications. Here are a few examples:

  1. Finance: Imagine you're comparing two different loan options. One loan might have a lower interest rate but higher fees, while the other has a higher interest rate but lower fees. You can set up a linear equation to figure out how much you would need to borrow for the total cost to be the same for both loans. This helps you make an informed decision about which loan is the better deal for you.

  2. Physics: Many physics problems involve linear relationships. For example, you might use a linear equation to calculate the distance an object travels at a constant speed over time. The equation d = vt (distance equals velocity times time) is a classic example of a linear equation used in physics.

  3. Chemistry: In chemistry, you might use linear equations to calculate the amount of a substance needed for a reaction or to determine the concentration of a solution. For instance, the ideal gas law (PV = nRT) can be rearranged into a linear form to solve for different variables.

  4. Everyday Life: Even in everyday situations, you use linear equations without even realizing it. For example, if you're trying to figure out how much to tip at a restaurant, you might calculate 15% or 20% of the bill. This is a linear calculation. Similarly, if you're figuring out how much gas money to split with friends on a road trip, you're using linear equations.

Practice Problems

To really master solving linear equations, practice is key! Here are a few problems you can try on your own:

  1. (3x + 2) / 4 = (2x - 1) / 3
  2. (5y - 3) / 2 = (4y + 1) / 5
  3. (7z + 4) / 6 = (5z - 2) / 3

Work through these problems using the steps we discussed earlier. Remember to eliminate the fractions, distribute, combine like terms, and isolate the variable. And don't forget to check your solutions!

Conclusion

So, we've successfully solved the equation (12y - 1)/2 = (9y + 8)/5 and found that y = 1/2. We walked through the solution step by step, explored alternative methods, discussed common mistakes to avoid, and even looked at real-world applications of linear equations. I hope you found this helpful and that you feel more confident in your ability to tackle similar problems in the future. Keep practicing, and you'll become a math whiz in no time!