Solving Equations Simplify 8(y-2)+3(4-y)=36 Step By Step

Hey guys! Let's dive into simplifying and solving this equation: 8(y-2) + 3(4-y) = 36. Equations can sometimes look intimidating, but don't worry, we'll break it down step by step, making it super easy to understand. Our goal here is to isolate 'y' on one side of the equation, which means we need to simplify the expressions on both sides first. Think of it like untangling a knot; we'll gently loosen each part until we have a clear, straight line. This involves using the distributive property, combining like terms, and then performing inverse operations. Remember, each step we take is a move towards revealing the value of 'y' that makes the equation true. So, grab your pencils, and let's get started on this mathematical adventure! We'll turn this complex-looking equation into a simple, solvable problem.

Step 1: Applying the Distributive Property

Okay, so our first mission is to tackle those parentheses in the equation 8(y-2) + 3(4-y) = 36. To do this, we're going to use something called the distributive property. It's a fancy name, but it's actually a pretty straightforward idea. Basically, it means we need to multiply the number outside the parentheses by each term inside the parentheses. Think of it like this: the number outside wants to 'share' with everyone inside! So, for the first part, 8(y-2), we'll multiply 8 by both 'y' and '-2'. That gives us 8 * y = 8y and 8 * -2 = -16. So, 8(y-2) becomes 8y - 16. Now, let's do the same for the second part, 3(4-y). We multiply 3 by both '4' and '-y'. That's 3 * 4 = 12 and 3 * -y = -3y. So, 3(4-y) becomes 12 - 3y. Remember, it's super important to pay attention to those signs – a negative sign can totally change the outcome! Now that we've distributed, our equation looks a little different, but we're one step closer to solving it. We've effectively 'opened up' the parentheses, which is a crucial step in simplifying any algebraic expression.

Step 2: Combining Like Terms

Alright, now that we've distributed and gotten rid of those parentheses, our equation looks like this: 8y - 16 + 12 - 3y = 36. The next step is to combine like terms. What exactly are 'like terms'? Well, they're terms that have the same variable raised to the same power – in our case, we have terms with 'y' and constant terms (just numbers without any variables). Think of it like sorting your laundry: you put all the shirts together, all the pants together, and so on. We're doing the same thing here! Let's start with the 'y' terms: we have 8y and -3y. If we combine them, we get 8y - 3y = 5y. It's like saying we have 8 'y's and we're taking away 3 'y's, leaving us with 5 'y's. Easy peasy! Now, let's combine the constant terms: we have -16 and +12. Combining these gives us -16 + 12 = -4. Think of it like owing 16 dollars and then paying back 12; you still owe 4 dollars. So, after combining like terms, our equation has slimmed down to 5y - 4 = 36. See how much simpler it looks now? We're making great progress in isolating 'y'.

Step 3: Isolating the Variable Term

Okay, we're getting closer to cracking this equation! We've simplified it to 5y - 4 = 36. Now, our mission is to isolate the variable term, which in this case is 5y. This means we want to get the 5y all by itself on one side of the equation. To do this, we need to get rid of that pesky -4 that's hanging out with it. How do we do that? By using the inverse operation. Remember, inverse operations are like opposites: addition is the opposite of subtraction, and multiplication is the opposite of division. Since we have 5y - 4, we need to do the opposite of subtracting 4, which is adding 4. But here's the golden rule of equations: whatever you do to one side, you must do to the other side. It's like a balancing scale; if you add something to one side, you have to add the same amount to the other side to keep it balanced. So, we'll add 4 to both sides of the equation: 5y - 4 + 4 = 36 + 4. On the left side, the -4 and +4 cancel each other out (they become zero!), leaving us with just 5y. On the right side, 36 + 4 equals 40. So, our equation now looks like 5y = 40. We've successfully isolated the variable term! We're just one step away from finding the value of 'y'.

Step 4: Solving for 'y'

We've reached the final stage, guys! Our equation is now 5y = 40. Remember our ultimate goal? It's to find the value of 'y'. Right now, 'y' is being multiplied by 5. So, to get 'y' all by itself, we need to do the inverse operation of multiplication, which is division. We're going to divide both sides of the equation by 5. This is like splitting a group of items into equal shares. So, we have 5y / 5 = 40 / 5. On the left side, the 5s cancel each other out, leaving us with just y. On the right side, 40 / 5 equals 8. So, we have y = 8. We did it! We've solved the equation! This means that the value of 'y' that makes the original equation true is 8. It's like we've found the missing piece of the puzzle. To be absolutely sure, we could plug 8 back into the original equation and see if it works. But for now, let's celebrate our success in simplifying and solving this equation. You've taken a complex-looking problem and broken it down into manageable steps. Great job!

Step 5: Verifying the Solution

Awesome! We've found that y = 8, but let's take it one step further and make absolutely sure our answer is correct. This is called verifying the solution, and it's a super important step in solving equations. Think of it like double-checking your work before you submit it. To verify our solution, we're going to plug y = 8 back into the original equation, which was 8(y-2) + 3(4-y) = 36. So, everywhere we see a 'y', we're going to replace it with '8'. This gives us 8(8-2) + 3(4-8) = 36. Now, we need to simplify both sides of the equation and see if they are equal. Let's start with the left side. Inside the first parentheses, we have 8 - 2 = 6, so 8(8-2) becomes 8 * 6 = 48. Inside the second parentheses, we have 4 - 8 = -4, so 3(4-8) becomes 3 * -4 = -12. Now our equation looks like 48 + (-12) = 36. Simplifying further, 48 + (-12) is the same as 48 - 12, which equals 36. So, the left side of the equation simplifies to 36. Now, let's look at the right side of the equation. It's already 36! So, we have 36 = 36. This is a true statement! Since both sides of the equation are equal when we substitute y = 8, we can be confident that our solution is correct. Verifying the solution is a fantastic way to build confidence in your answer and ensure you haven't made any mistakes along the way. Pat yourself on the back; you've not only solved the equation but also confirmed your solution!

Conclusion: Mastering Equation Simplification

Alright, guys, we've taken on the equation 8(y-2) + 3(4-y) = 36 and conquered it! We've walked through each step, from applying the distributive property to combining like terms, isolating the variable, solving for 'y', and even verifying our solution. It's like we've climbed a mathematical mountain and reached the summit! Solving equations is a fundamental skill in algebra, and it's something you'll use again and again in mathematics and beyond. The key takeaway here is that complex equations can be broken down into simpler, manageable steps. Each step we took had a clear purpose: to simplify the equation and get closer to isolating 'y'. Remember, the distributive property helps us get rid of parentheses, combining like terms makes the equation less cluttered, and inverse operations allow us to isolate the variable. And don't forget the importance of verifying your solution – it's like the final stamp of approval on your work. By mastering these steps, you'll be able to tackle a wide range of algebraic equations with confidence. Keep practicing, and you'll become an equation-solving pro in no time! You've got this!