Solving For Y: A Step-by-Step Guide

by Andrew McMorgan 36 views

Hey guys! Ever find yourself staring at an equation with a lone 'y' and a bunch of fractions, wondering how to isolate that variable? You're definitely not alone! Today, we're going to break down a common algebraic problem step by step: solving for 'y' in the equation (7/4)y - 8/5 = -2/3. Don't worry, it's not as scary as it looks. We'll walk through each step together, making sure you understand the why behind the how. So, grab your pencils and let's dive in!

Understanding the Equation

Before we start crunching numbers, let's make sure we understand what the equation is telling us. The equation (7/4)y - 8/5 = -2/3 is a linear equation, meaning it represents a straight line when graphed. Our goal is to find the value of 'y' that makes this equation true. Think of it like a puzzle where we need to find the missing piece. The 'y' is currently being multiplied by 7/4, and then 8/5 is being subtracted from the result. The entire expression is equal to -2/3. To solve for 'y', we need to undo these operations in the reverse order. This is a fundamental concept in algebra: we use inverse operations to isolate the variable we're trying to solve for. For example, the inverse operation of addition is subtraction, and the inverse operation of multiplication is division. We'll be using these concepts throughout the solution, so make sure you're comfortable with them. Remember, the key is to maintain the balance of the equation. Whatever operation we perform on one side, we must also perform on the other side to keep the equation true. This is like a scale – if we add weight to one side, we need to add the same weight to the other side to keep it balanced.

Step 1: Isolating the Term with 'y'

Our first mission is to isolate the term that contains 'y', which is (7/4)y. To do this, we need to get rid of the -8/5 that's being subtracted from it. Remember what we talked about earlier? We use the inverse operation! The inverse of subtraction is addition, so we'll add 8/5 to both sides of the equation. This is a crucial step because it keeps the equation balanced. We're essentially moving the -8/5 to the other side of the equation by adding its inverse. Let's write it out: (7/4)y - 8/5 + 8/5 = -2/3 + 8/5. On the left side, -8/5 and +8/5 cancel each other out, leaving us with just (7/4)y. Now, we need to deal with the right side of the equation: -2/3 + 8/5. We're adding two fractions with different denominators, so we need to find a common denominator. The least common multiple of 3 and 5 is 15. So, we'll rewrite -2/3 as -10/15 and 8/5 as 24/15. Now we can add them: -10/15 + 24/15 = 14/15. So, our equation now looks like this: (7/4)y = 14/15. We've successfully isolated the term with 'y' on one side of the equation. We're one step closer to finding the value of 'y'!

Step 2: Getting 'y' by Itself

Awesome! We've got (7/4)y = 14/15. Now, we need to get 'y' completely alone. Currently, 'y' is being multiplied by 7/4. To undo this multiplication, we use the inverse operation: division. But dividing by a fraction can be a bit tricky, so we'll use a handy shortcut: multiplying by the reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. So, the reciprocal of 7/4 is 4/7. We'll multiply both sides of the equation by 4/7. This is another crucial step in maintaining the balance of the equation. Whatever we do to one side, we must do to the other. Let's write it out: (4/7) * (7/4)y = (4/7) * (14/15). On the left side, (4/7) * (7/4) cancels out, leaving us with just 'y'. That's exactly what we wanted! On the right side, we have (4/7) * (14/15). Before we multiply, let's see if we can simplify anything. We notice that 7 and 14 have a common factor of 7. We can divide both 7 and 14 by 7, which gives us 1 and 2, respectively. Now our multiplication looks like this: (4/1) * (2/15). Multiplying the numerators, we get 4 * 2 = 8. Multiplying the denominators, we get 1 * 15 = 15. So, the right side simplifies to 8/15. Our equation now looks like this: y = 8/15. We've done it! We've successfully solved for 'y'!

Step 3: Verifying the Solution

Before we celebrate too much, let's make sure our answer is correct. It's always a good idea to verify your solution in math problems. To do this, we'll substitute our value of 'y' (8/15) back into the original equation: (7/4)y - 8/5 = -2/3. Let's plug it in: (7/4) * (8/15) - 8/5 = -2/3. Now we need to simplify. First, let's multiply (7/4) * (8/15). We can simplify before multiplying by noticing that 4 and 8 have a common factor of 4. Dividing both by 4, we get 1 and 2, respectively. So, the multiplication becomes (7/1) * (2/15) = 14/15. Now our equation looks like this: 14/15 - 8/5 = -2/3. We need to subtract 8/5 from 14/15. To do this, we need a common denominator. The least common multiple of 15 and 5 is 15. So, we'll rewrite 8/5 as 24/15. Now we can subtract: 14/15 - 24/15 = -10/15. We can simplify -10/15 by dividing both the numerator and the denominator by their greatest common factor, which is 5. This gives us -2/3. So, our equation becomes: -2/3 = -2/3. Hooray! The left side equals the right side, which means our solution is correct. We've successfully verified that y = 8/15 is the solution to the equation.

Conclusion: You Did It!

Alright, guys, give yourselves a pat on the back! You've tackled a potentially tricky equation and come out on top. Solving for 'y' in the equation (7/4)y - 8/5 = -2/3 might have seemed daunting at first, but by breaking it down into manageable steps, we were able to find the solution: y = 8/15. Remember, the key to solving algebraic equations is to understand the underlying principles, use inverse operations, and keep the equation balanced. And don't forget to verify your solution! Keep practicing, and you'll become a math whiz in no time. Now, go forth and conquer more equations! You got this!

Practice Problems

Want to test your skills further? Try solving these equations for 'y':

  1. (3/2)y + 1/4 = 5/6
  2. (5/3)y - 2/7 = 1/2
  3. (1/5)y + 4/9 = -2/3

Share your answers in the comments below! We'd love to see how you're doing.

Further Learning

If you're interested in learning more about solving linear equations, check out these resources:

Keep exploring and keep learning!