Solving System Of Equations: A Step-by-Step Guide

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Hey guys! Ever found yourself staring at a bunch of equations and feeling totally lost? Don't worry, you're not alone! Solving systems of equations can seem tricky at first, but with the right approach, it becomes a piece of cake. This guide will walk you through solving the system:

  • 20x + y - 27 = 17
  • 3x + 20y - 7 = -28
  • 2x - 3y + 207 = 25

We'll break it down step-by-step, so you'll be a pro in no time! So, let’s dive in and make math fun!

Understanding Systems of Equations

Before we jump into solving this specific system, let's make sure we're all on the same page about what a system of equations actually is. Essentially, it's a set of two or more equations that involve the same variables. Our goal is to find values for these variables that satisfy all equations simultaneously. Think of it like finding the perfect combination that makes everything true!

In our case, we have three equations with two variables, x and y. This means we're looking for a single pair of (x, y) values that work for all three equations. It might seem a bit daunting, but we've got this! The key here is to understand that each equation represents a relationship between x and y, and the solution we seek is the point where these relationships intersect.

Why are Systems of Equations Important?

Now, you might be wondering, why bother learning this stuff? Well, systems of equations pop up everywhere in real life! From calculating mixtures in chemistry to optimizing business strategies, these mathematical tools are incredibly versatile. For example, imagine you're trying to figure out the right blend of ingredients for a recipe or determining the optimal pricing for your products. Systems of equations can help you model these situations and find the best solutions. By grasping the fundamentals, you're not just acing math problems; you're also developing critical thinking skills that will benefit you in countless ways. So, stick with us, and you'll see how powerful this knowledge can be!

Simplifying the Equations

Okay, first things first, let's make our equations a little easier to work with. We want to get all the constants (the numbers without variables) on one side of the equation. This will help us see the relationships between x and y more clearly. Remember, the goal here is to tidy things up so we can tackle the problem head-on. Think of it like organizing your workspace before starting a big project – a little preparation goes a long way!

Let's start with the first equation:

  • 20x + y - 27 = 17

To isolate the variables, we'll add 27 to both sides of the equation:

  • 20x + y = 17 + 27
  • 20x + y = 44

Great! Now, let's move on to the second equation:

  • 3x + 20y - 7 = -28

We'll add 7 to both sides:

  • 3x + 20y = -28 + 7
  • 3x + 20y = -21

And finally, the third equation:

  • 2x - 3y + 207 = 25

Subtract 207 from both sides:

  • 2x - 3y = 25 - 207
  • 2x - 3y = -182

So, our simplified system of equations looks like this:

  • 20x + y = 44
  • 3x + 20y = -21
  • 2x - 3y = -182

See? Much cleaner! Now we have a clear view of the relationships between x and y. This is a crucial step because it sets the stage for the next phase: choosing a method to solve the system.

Choosing a Method: Elimination or Substitution?

Now that our equations are nice and tidy, we need to decide how we're going to solve them. There are two main methods for tackling systems of equations: elimination and substitution. Both are powerful tools, but sometimes one method is more efficient than the other. It's like having a toolbox full of different wrenches – you want to pick the one that fits the nut best!

The elimination method involves manipulating the equations so that when you add them together, one of the variables cancels out. This leaves you with a single equation in one variable, which is much easier to solve. The idea is to strategically eliminate one variable, making the problem simpler.

The substitution method, on the other hand, involves solving one equation for one variable and then substituting that expression into another equation. This also results in a single equation in one variable. Think of it as replacing one piece of the puzzle with another that represents the same thing.

In our case, let's consider the equations we have:

  • 20x + y = 44
  • 3x + 20y = -21
  • 2x - 3y = -182

Looking at these, the elimination method might be a bit tricky because it's not immediately obvious how to make coefficients match up easily. However, notice that the first equation has a single 'y' term. This makes it a good candidate for solving for 'y' and then using substitution. So, for this particular system, we'll go with the substitution method. But remember, both methods are valid, and the best choice often depends on the specific equations you're dealing with.

Solving for y in the First Equation

Alright, team! We've decided to use the substitution method, and the first step is to solve one of our equations for one of the variables. As we discussed, the first equation, 20x + y = 44, looks like a prime candidate for solving for y. It’s already pretty close to being in the form we want!

To isolate y, we need to get rid of the 20x term. We can do this by subtracting 20x from both sides of the equation. Remember, whatever we do to one side of the equation, we have to do to the other to keep things balanced. It’s like a mathematical seesaw – we want to keep it level!

So, we have:

  • 20x + y - 20x = 44 - 20x

Simplifying this gives us:

  • y = 44 - 20x

Fantastic! We've now solved the first equation for y. This expression, y = 44 - 20x, tells us how y is related to x in this particular equation. This is a crucial piece of information that we're going to use in the next step. Think of it as finding a key ingredient for our mathematical recipe. Now, we're ready to substitute this expression into another equation to eliminate y and solve for x.

Substituting into the Second Equation

Okay, we've got y = 44 - 20x. Now comes the fun part: substitution! We're going to take this expression for y and plug it into one of the other equations. We can't use the first equation again (that's where we got the expression for y), so let's use the second equation: 3x + 20y = -21.

Wherever we see a 'y' in the second equation, we're going to replace it with '(44 - 20x)'. It's like we're swapping out one piece of the puzzle for another that we know is equivalent. This might look a little messy at first, but trust the process! It's all about carefully replacing the variable to create a new equation in just one variable.

So, the second equation becomes:

  • 3x + 20(44 - 20x) = -21

Now, we have an equation with only x in it. This is exactly what we wanted! We've eliminated y from the equation, making it much easier to solve. The next step is to simplify this equation and solve for x. We'll need to distribute, combine like terms, and isolate x. But don't worry, we'll take it one step at a time. This is where the real problem-solving begins, and you're doing great so far!

Simplifying and Solving for x

Alright, let's tackle that equation we got after the substitution: 3x + 20(44 - 20x) = -21. The first thing we need to do is distribute the 20 across the terms inside the parentheses. Remember the distributive property? It's like sharing the wealth – the 20 needs to multiply both the 44 and the -20x.

So, we get:

  • 3x + 880 - 400x = -21

Now, let's combine the like terms on the left side of the equation. We have two terms with 'x' in them (3x and -400x) and a constant term (880). Combining the 'x' terms, we get:

  • -397x + 880 = -21

Next, we want to isolate the 'x' term, so we'll subtract 880 from both sides of the equation:

  • -397x = -21 - 880
  • -397x = -901

Finally, to solve for x, we'll divide both sides by -397:

  • x = -901 / -397
  • x ≈ 2.27

Woohoo! We've found the value of x. It's approximately 2.27. This is a major milestone in solving our system of equations. But remember, we're not done yet. We still need to find the value of y. Now that we have x, we can plug it back into one of our equations to solve for y. Keep up the awesome work!

Finding the Value of y

Excellent! We've successfully found x, which is approximately 2.27. Now, it's time to find y. Remember that expression we found earlier, y = 44 - 20x? This is going to be our best friend right now. We can simply substitute the value of x we just found into this equation to solve for y.

So, let's plug in x ≈ 2.27:

  • y = 44 - 20(2.27)

Now, we just need to do the arithmetic. First, multiply 20 by 2.27:

  • y = 44 - 45.4

Then, subtract 45.4 from 44:

  • y = -1.4

Great! We've found y, which is -1.4. So, we have a potential solution: x ≈ 2.27 and y = -1.4. But before we declare victory, there's one crucial step we need to take: checking our solution.

Checking the Solution

Okay, we've found x and y, but we're not quite done yet! It's super important to check our solution to make sure it actually works. Think of it like double-checking your work on a test – you want to catch any mistakes before you submit your answer. In this case, we need to make sure that our values for x and y satisfy all three of our original equations.

Let's start with the first equation: 20x + y = 44. We'll plug in x ≈ 2.27 and y = -1.4:

  • 20(2.27) + (-1.4) ≈ 44
  • 45.4 - 1.4 ≈ 44
  • 44 ≈ 44

So far, so good! The first equation checks out. Now, let's move on to the second equation: 3x + 20y = -21:

  • 3(2.27) + 20(-1.4) ≈ -21
  • 6.81 - 28 ≈ -21
  • -21.19 ≈ -21

This is pretty close! The slight difference might be due to rounding our value for x. Let's check the third equation: 2x - 3y = -182:

  • 2(2.27) - 3(-1.4) ≈ -182
  • 4.54 + 4.2 ≈ -182
  • 8.74 ≈ -182

Oops! It seems like this equation is not satisfied by our values of x and y. This indicates there may be an error in our calculations, or the system of equations may not have a common solution that satisfies all three equations simultaneously. Since we have three equations and only two variables, it's possible that the system is overdetermined and inconsistent.

Conclusion: Inconsistent System

Alright, guys, we've gone through the entire process of solving a system of equations, and it turns out this one is a bit of a special case! We meticulously simplified the equations, chose the substitution method, solved for x and y, and then checked our solution. Unfortunately, our check revealed that the values we found for x and y satisfy the first two equations but not the third.

This means that there is no single pair of (x, y) values that will make all three equations true at the same time. In mathematical terms, we say that this system of equations is inconsistent. This can happen when you have more equations than variables, as we do in this case. Think of it like trying to fit a square peg into a round hole – sometimes, things just don't line up perfectly.

So, while we didn't find a single solution that works for all equations, we learned a valuable lesson about the nature of systems of equations. Not every system has a solution, and it's important to check your work to make sure your answers are valid. Don't be discouraged if you encounter an inconsistent system – it's just part of the mathematical landscape! Keep practicing, and you'll become a master problem-solver in no time.