Solving For Y: Y - 5/7 * Y = 2/9? A Math Guide

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Hey guys! Let's break down this math problem together. If you're scratching your head over the equation y - 5/7 * y = 2/9, you've come to the right place. We're going to dive deep into solving this equation, making sure every step is crystal clear. Math can seem daunting sometimes, but trust me, with a bit of patience and the right approach, you'll be acing these problems in no time. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into the solution, let's make sure we fully understand the equation we're dealing with: y - 5/7 * y = 2/9. This equation involves a variable, y, and our goal is to isolate this variable to find its value. To do this effectively, we need to follow the order of operations (PEMDAS/BODMAS) and apply some algebraic principles.

First off, remember the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is super important because it dictates the sequence in which we perform calculations.

In our equation, we have subtraction and multiplication. According to the order of operations, we need to handle the multiplication before the subtraction. This means we'll first deal with 5/7 * y. Think of y as 1 * y, which will help us combine like terms later on. This initial understanding is key because it lays the groundwork for the rest of the solution. If you grasp this part, the rest will flow much smoother, I promise!

Combining Like Terms

The heart of solving this equation lies in combining like terms. What are like terms, you ask? Well, they are terms that have the same variable raised to the same power. In our case, we have two terms involving y: y (which is really 1*y) and -5/7 * y. We can combine these because they both have y to the power of 1.

To combine them, we need to think about the coefficients (the numbers in front of the y). We have 1 (from the y) and -5/7. So, we're essentially doing the subtraction 1 - 5/7. To subtract these, we need a common denominator. Remember fractions? This is where they come into play! We can rewrite 1 as 7/7. Now we have 7/7 - 5/7.

Subtracting the numerators (the top numbers) while keeping the denominator (the bottom number) the same, we get (7 - 5) / 7, which simplifies to 2/7. So, when we combine the y terms, y - 5/7 * y becomes 2/7 * y. See? We're making progress! This step is crucial because it simplifies our equation, bringing us closer to isolating y and finding its value. By understanding how to combine like terms, you’re building a solid foundation for tackling more complex algebraic equations.

Isolating the Variable

Now that we've combined like terms, our equation looks much simpler: 2/7 * y = 2/9. The next crucial step is isolating the variable, which means getting y all by itself on one side of the equation. To do this, we need to undo the operation that's currently being applied to y. In this case, y is being multiplied by 2/7.

The golden rule of algebra is that whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced. To undo multiplication, we use division. So, we need to divide both sides of the equation by 2/7. But here's a little trick: dividing by a fraction is the same as multiplying by its reciprocal.

The reciprocal of a fraction is simply flipping it over. So, the reciprocal of 2/7 is 7/2. Now, instead of dividing by 2/7, we can multiply both sides of the equation by 7/2. This looks like: (2/7 * y) * (7/2) = (2/9) * (7/2). On the left side, the 2/7 and 7/2 cancel each other out, leaving us with just y, which is exactly what we wanted!

On the right side, we multiply 2/9 by 7/2. Multiplying fractions is straightforward: you multiply the numerators together and the denominators together. So, (2/9) * (7/2) becomes (2 * 7) / (9 * 2), which equals 14/18. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 14/18 simplifies to 7/9. And there you have it: y = 7/9. We've successfully isolated the variable and found its value!

Step-by-Step Solution

Let's recap the entire process with a clear, step-by-step solution. This will help solidify your understanding and give you a handy reference for tackling similar problems in the future. Breaking it down like this makes it super easy to follow, so you can nail it every time.

  1. Original Equation: Start with the equation we're trying to solve: y - 5/7 * y = 2/9.
  2. Combine Like Terms: We have two terms with y. First, recognize that y is the same as 1 * y. So, we have 1 * y - 5/7 * y. To combine these, we need a common denominator. Convert 1 to 7/7, so we have 7/7 * y - 5/7 * y. Subtracting the fractions gives us (7/7 - 5/7) * y = 2/7 * y.
  3. Simplified Equation: After combining like terms, our equation is now: 2/7 * y = 2/9.
  4. Isolate the Variable: To get y by itself, we need to undo the multiplication by 2/7. We do this by multiplying both sides of the equation by the reciprocal of 2/7, which is 7/2. So, we have (2/7 * y) * (7/2) = (2/9) * (7/2).
  5. Multiply by the Reciprocal: On the left side, 2/7 and 7/2 cancel each other out, leaving us with just y. On the right side, we multiply the fractions: (2 * 7) / (9 * 2) = 14/18.
  6. Simplify the Fraction: We can simplify 14/18 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 14/18 = 7/9.
  7. Solution: Therefore, the solution to the equation is y = 7/9.

See? Breaking it down step-by-step makes it super clear. You start with the original equation, combine those like terms to simplify things, and then isolate the variable by performing the inverse operation. It's all about keeping the equation balanced and following those algebraic rules.

Common Mistakes to Avoid

Alright, guys, let’s talk about some common pitfalls that students often stumble upon when solving equations like this. Knowing these mistakes ahead of time can seriously boost your accuracy and confidence. Trust me, spotting these errors early can save you a lot of headaches!

  • Forgetting the Order of Operations: This is a biggie! As we discussed earlier, the order of operations (PEMDAS/BODMAS) is crucial. A common mistake is subtracting 5/7 from 1 before multiplying by y. Remember, multiplication comes before subtraction. So always handle that 5/7 * y first.
  • Incorrectly Combining Like Terms: When combining like terms, make sure you're only adding or subtracting terms that have the same variable raised to the same power. In our equation, it's tempting to just subtract 5/7 from 1 without properly accounting for the y. Always keep those variables in mind!
  • Not Using a Common Denominator: This is a classic fraction fumble. When adding or subtracting fractions, you absolutely need a common denominator. If you forget this, your calculations will be off. Remember to convert whole numbers into fractions with the same denominator before combining.
  • Dividing Instead of Multiplying by the Reciprocal: When isolating the variable, we multiply by the reciprocal of the fraction. Some folks mistakenly divide by the fraction, which can lead to a whole different answer. Remember, dividing by a fraction is the same as multiplying by its reciprocal – that flip is key!
  • Not Simplifying Fractions: Always, always, always simplify your fractions to their lowest terms. It’s not only good practice, but it also makes your final answer cleaner and easier to work with. If you end up with something like 14/18, take that extra step to reduce it to 7/9.
  • Forgetting to Apply the Operation to Both Sides: This is the golden rule of algebra! Whatever you do to one side of the equation, you must do to the other. If you only multiply one side by the reciprocal, your equation becomes unbalanced, and your answer will be wrong. Keep that balance in check!

By being aware of these common mistakes, you can actively avoid them. Double-check your work, especially at these critical steps, and you’ll be solving equations like a pro in no time!

Practice Problems

Okay, now that we've walked through the solution and covered the common pitfalls, it's time to put your skills to the test! Practice makes perfect, right? So, let's dive into some practice problems that are similar to the one we just solved. Working through these will help you solidify your understanding and build confidence. Grab your pencil and paper, and let's get to it!

Here are a few equations you can try:

  1. z - 2/3 * z = 4/5
  2. x - 3/4 * x = 1/2
  3. a - 1/5 * a = 3/10
  4. b - 4/9 * b = 2/3
  5. c - 5/8 * c = 1/4

For each of these problems, follow the same steps we outlined earlier:

  • Combine Like Terms: Identify the terms with the variable and combine them.
  • Isolate the Variable: Use the reciprocal to get the variable by itself on one side of the equation.
  • Simplify: Reduce any fractions to their simplest form.

Don't just rush through the problems; take your time and focus on understanding each step. If you get stuck, go back to the step-by-step solution we worked through earlier. Remember, the goal isn't just to get the right answer, but to understand the process.

After you’ve solved these, you can even create your own practice problems by changing the fractions and constants. This is a fantastic way to challenge yourself and deepen your understanding. Solving math problems is like building a muscle – the more you use it, the stronger it gets!

So, give these practice problems a try, and watch your equation-solving skills soar!

Conclusion

Alright, guys! We've reached the end of our journey to solve the equation y - 5/7 * y = 2/9. We've covered a lot of ground, from understanding the equation and combining like terms to isolating the variable and avoiding common mistakes. You’ve armed yourselves with some serious math skills today!

Remember, the key to mastering these types of problems is practice and patience. Don't get discouraged if you stumble along the way. Math is like learning a new language – it takes time and effort. The more you practice, the more fluent you'll become.

We started by breaking down the original equation, emphasizing the importance of the order of operations. Then, we tackled combining like terms, a crucial step in simplifying the equation. Isolating the variable was our next big challenge, and we conquered it by multiplying both sides of the equation by the reciprocal.

We also highlighted some common mistakes to watch out for, like forgetting the order of operations or not using a common denominator. By being aware of these pitfalls, you can steer clear of them and boost your accuracy.

Finally, we provided you with a set of practice problems to test your skills. Practice is the name of the game, and the more you solve, the more confident you'll become. So, keep those pencils moving and those brains buzzing!

So, whether you're tackling homework, studying for a test, or just brushing up on your math skills, remember the strategies and steps we've discussed. Keep practicing, stay patient, and you'll be solving equations like a math whiz in no time. You've got this! High five!