Solving The System Of Equations 8x-y=3x+3y=16 A Comprehensive Guide

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Hey guys! Ever found yourself scratching your head over a math problem that seems like it’s speaking another language? Well, you're definitely not alone! Today, we're going to tackle a type of problem that often pops up in algebra: solving systems of linear equations. Specifically, we're diving deep into finding the solutions for a system that looks like this: 8x - y = 3x + 3y = 16. Sounds intimidating? Don't worry, we'll break it down step by step so that by the end of this guide, you'll be solving these equations like a pro. Let’s jump right in!

Understanding Systems of Linear Equations

Before we even think about diving into the nitty-gritty of solving linear equations, let's take a step back and really understand what we're dealing with. Imagine you have two or more equations, each representing a straight line when graphed. A system of linear equations is simply a collection of these equations considered together. The solution to this system? It's the point (or points) where these lines intersect. Think of it as the sweet spot that satisfies all the equations at the same time. Now, why is this important? Well, systems of linear equations pop up everywhere in real life, from figuring out the cost of items at a store to more complex scenarios in engineering and economics. So, mastering this skill is super practical!

In our specific case, we're faced with 8x - y = 3x + 3y = 16. This might look a bit unusual because it seems like we have three expressions all equal to each other. But don't let that throw you off! We can simply break this down into two separate equations. The first equation comes from equating the first two expressions: 8x - y = 3x + 3y. And the second equation? That's from equating the second expression to the constant: 3x + 3y = 16. See? We've now transformed our initial problem into a more standard system of two equations, which we can then solve using various methods. We’re essentially looking for the values of x and y that make both of these equations true simultaneously. This is crucial, as it highlights the core concept of solving systems of equations: finding the common ground between multiple conditions.

Breaking Down the Problem: Separating the Equations

Okay, let's get down to the nitty-gritty. Remember how we talked about our original equation, 8x - y = 3x + 3y = 16, being a bit of a triple threat? Well, we're going to tame it by splitting it into two manageable equations. This is a crucial first step because it transforms a seemingly complex problem into something we can actually work with. Think of it like untangling a knot – once you find the right starting point, everything else becomes much easier.

So, how do we do this exactly? We're going to use the fact that if a = b = c, then it must be true that a = b and b = c. Simple, right? Applying this to our equation, we can say:

  1. Equation 1: 8x - y = 3x + 3y
  2. Equation 2: 3x + 3y = 16

Now we have two equations, each giving us a different piece of the puzzle. Equation 1 relates x and y in a certain way, while Equation 2 gives us another relationship between them. To solve for x and y, we need to use both equations together. It’s like having two clues to the same mystery – neither clue alone is enough, but together they can lead us to the solution. By breaking down the problem in this way, we’ve set the stage for using methods like substitution or elimination, which we'll dive into next. Trust me, guys, this step is super important. It’s all about making the complex simple!

Method 1: The Substitution Method

Alright, let's talk about one of the coolest techniques in our equation-solving toolkit: the substitution method. This method is like being a detective – we're going to solve for one variable in terms of the other and then substitute that expression into the other equation. It might sound a bit like a mouthful, but trust me, it's simpler than it sounds! The beauty of the substitution method is its versatility; it’s particularly handy when one of the equations can be easily solved for one variable. So, let's see how this works with our system:

  1. 8x - y = 3x + 3y
  2. 3x + 3y = 16

Our first step is to pick an equation and solve for one variable. Looking at Equation 1, it seems easiest to isolate y. Let’s rearrange it:

8x - y = 3x + 3y 8x - 3x = 3y + y 5x = 4y y = (5/4)x

See what we did there? We've expressed y in terms of x. Now comes the fun part – the substitution! We're going to take this expression for y and plug it into Equation 2:

3x + 3y = 16 3x + 3((5/4)x) = 16 3x + (15/4)x = 16

Now we have an equation with just one variable, x. This is great because we can solve for x directly. Let’s simplify:

(12/4)x + (15/4)x = 16 (27/4)x = 16 x = (16 * 4) / 27 x = 64/27

We've found x! Now, to find y, we simply substitute this value of x back into our expression for y:

y = (5/4)x y = (5/4) * (64/27) y = (5 * 16) / 27 y = 80/27

And there you have it! Our solution is x = 64/27 and y = 80/27. The substitution method is like a puzzle – you find one piece and then use it to find the others. It's all about strategically replacing one variable to simplify the equation. Cool, right?

Method 2: The Elimination Method

Okay, guys, let's switch gears and explore another fantastic method for solving systems of linear equations: the elimination method! This technique is super powerful, especially when our equations are set up in a way that makes it easy to cancel out one of the variables. Think of it as a strategic subtraction game – we're going to manipulate our equations so that when we add or subtract them, one variable disappears, leaving us with a simpler equation to solve. So, how does this work with our trusty system?

  1. 8x - y = 3x + 3y
  2. 3x + 3y = 16

First things first, we need to rewrite Equation 1 to make it look a bit more like Equation 2. Let's get all the variables on one side:

8x - y = 3x + 3y 8x - 3x = 3y + y 5x = 4y 5x - 4y = 0

Now our system looks like this:

  1. 5x - 4y = 0
  2. 3x + 3y = 16

The goal of the elimination method is to make the coefficients of either x or y the same (but with opposite signs) in both equations. This way, when we add the equations, that variable will be eliminated. Looking at our equations, it seems easiest to eliminate y. To do this, we can multiply Equation 3 by 3 and Equation 4 by 4:

(5x - 4y) * 3 = 0 * 3 --> 15x - 12y = 0 (3x + 3y) * 4 = 16 * 4 --> 12x + 12y = 64

Now we have:

  1. 15x - 12y = 0
  2. 12x + 12y = 64

See how the y terms have the same coefficient but opposite signs? This is perfect! Now, let’s add the equations:

(15x - 12y) + (12x + 12y) = 0 + 64 27x = 64 x = 64/27

Just like with the substitution method, we've found x! Now we need to find y. We can substitute this value of x into either Equation 3 or Equation 4. Let’s use Equation 4:

3x + 3y = 16 3(64/27) + 3y = 16 (64/9) + 3y = 16 3y = 16 - (64/9) 3y = (144 - 64) / 9 3y = 80/9 y = (80/9) / 3 y = 80/27

And there we have it! Our solution, once again, is x = 64/27 and y = 80/27. The elimination method is like a strategic puzzle where we manipulate equations to make a variable vanish. It's super satisfying when you see those terms cancel out, isn't it?

Verifying the Solution

Alright, we've crunched the numbers and found our solution: x = 64/27 and y = 80/27. But before we pat ourselves on the back, there's a crucial step we absolutely cannot skip: verifying the solution! Think of this as the final check, the detective making sure all the pieces fit. It's super important because it ensures that our values for x and y actually work in the original equations. There’s nothing worse than thinking you’ve solved a problem, only to find out your answer doesn’t quite hold up.

So, how do we verify? Simple! We're going to plug our values for x and y back into our original equations and see if they hold true. Remember our system?

  1. 8x - y = 3x + 3y
  2. 3x + 3y = 16

Let's start with Equation 1:

8x - y = 3x + 3y 8(64/27) - (80/27) = 3(64/27) + 3(80/27) (512/27) - (80/27) = (192/27) + (240/27) 432/27 = 432/27

Awesome! The left side equals the right side, so our solution works for Equation 1. Now, let's check Equation 2:

3x + 3y = 16 3(64/27) + 3(80/27) = 16 (192/27) + (240/27) = 16 432/27 = 16 16 = 16

Woo-hoo! It works for Equation 2 as well! This means our solution x = 64/27 and y = 80/27 is the real deal. Verifying the solution is like the ultimate seal of approval. It gives us the confidence to say,