Solving Systems Of Equations A Detailed Guide

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Hey guys! Today, we're diving deep into the fascinating world of solving systems of equations. Specifically, we're going to tackle a system with three equations and three unknowns. Don't worry if it sounds intimidating; we'll break it down step by step, making it super easy to follow. Whether you're a student brushing up on your algebra skills or just someone who loves a good math puzzle, this guide is for you. We’ll be focusing on a particular system, but the techniques we learn can be applied to many similar problems. So, grab your pencils, and let's get started!

Understanding Systems of Equations

Before we jump into the nitty-gritty of our specific problem, let's take a moment to understand what a system of equations actually is. Simply put, a system of equations is a set of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all the equations simultaneously. Think of it as a puzzle where each equation gives you a piece of the solution, and you need to fit all the pieces together.

In our case, we have a system of three linear equations:

  1. x + y + z = 120
  2. x - 2y + z = 0
  3. 2x + 2y - z = 75

Each of these equations represents a plane in three-dimensional space. The solution to the system is the point (or set of points) where all three planes intersect. This intersection point gives us the values for x, y, and z that make all three equations true. Solving such systems is a cornerstone of algebra and has applications in various fields, including engineering, physics, and economics. So, it’s definitely a skill worth mastering! Now that we’ve got a handle on the basics, let’s dive into how to actually solve this particular system.

Our System of Equations

Let’s reiterate the system we’re about to solve so it’s fresh in our minds. We have the following three equations:

  1. x + y + z = 120
  2. x - 2y + z = 0
  3. 2x + 2y - z = 75

We're looking for values for x, y, and z that will make all three of these equations true at the same time. There are several methods we can use to solve a system like this, including substitution, elimination, and matrix methods. Today, we'll focus on the elimination method because it's particularly efficient for systems with multiple variables. The elimination method involves strategically adding or subtracting multiples of equations to eliminate one variable at a time, making the system simpler to solve. This method is not only powerful but also provides a clear, step-by-step approach to solving complex systems. By mastering the elimination method, you'll be equipped to tackle a wide range of algebraic challenges. So, with our system clearly defined and our method chosen, let’s move on to the first step: eliminating a variable!

The Elimination Method: Step-by-Step

The elimination method is all about strategically combining equations to get rid of one variable at a time. This makes the system simpler to solve. We'll walk through each step, so you can see exactly how it works.

Step 1: Eliminate x from the Second Equation

Our first goal is to eliminate x from the second equation. We can do this by subtracting the second equation from the first equation. Let's label our equations for clarity:

  1. x + y + z = 120 (Equation 1)
  2. x - 2y + z = 0 (Equation 2)
  3. 2x + 2y - z = 75 (Equation 3)

To eliminate x, we subtract Equation 2 from Equation 1:

(x + y + z) - (x - 2y + z) = 120 - 0

Simplifying this, we get:

3y = 120

Now, we can solve for y:

y = 120 / 3

y = 40

Great! We've found the value of y. This is a significant step forward. But we're not done yet. We still need to find x and z. Our next step will be to use this value of y to eliminate x from another equation.

Step 2: Eliminate x from the Third Equation

Now that we know y = 40, let's use that information to eliminate x from the third equation. We'll use Equation 1 and Equation 3 again.

  1. x + y + z = 120
  2. 2x + 2y - z = 75

To eliminate x, we can multiply Equation 1 by 2 and then subtract Equation 3 from the result. This gives us:

2 * (x + y + z) = 2 * 120

2x + 2y + 2z = 240

Now, subtract Equation 3 from this new equation:

(2x + 2y + 2z) - (2x + 2y - z) = 240 - 75

Simplifying, we get:

3z = 165

Now, we can solve for z:

z = 165 / 3

z = 55

Fantastic! We've found the value of z. We’re on a roll! We now know y = 40 and z = 55. All that’s left is to find x. We’re in the home stretch now!

Step 3: Solve for x

We've found y = 40 and z = 55. Now we just need to find x. We can use any of our original equations to do this. Let's use Equation 1 because it looks the simplest:

x + y + z = 120

Substitute the values of y and z:

x + 40 + 55 = 120

Simplify:

x + 95 = 120

Now, solve for x:

x = 120 - 95

x = 25

We did it! We've found the value of x. We now have all the pieces of the puzzle. Let's summarize our findings.

The Solution

After all our hard work, we've found the values for x, y, and z that satisfy our system of equations. Let's recap our results:

  • x = 25
  • y = 40
  • z = 55

So, the solution to the system is the ordered triple (25, 40, 55). This means that when x is 25, y is 40, and z is 55, all three of our original equations are true. But before we celebrate too much, it's always a good idea to check our work. Let's plug these values back into the original equations to make sure they hold true.

Checking Our Solution

It's crucial to verify our solution to ensure we haven't made any mistakes along the way. Let's plug our values x = 25, y = 40, and z = 55 into each of the original equations:

  1. x + y + z = 120

    25 + 40 + 55 = 120

    120 = 120 (Correct!)

  2. x - 2y + z = 0

    25 - 2(40) + 55 = 0

    25 - 80 + 55 = 0

    0 = 0 (Correct!)

  3. 2x + 2y - z = 75

    2(25) + 2(40) - 55 = 75

    50 + 80 - 55 = 75

    75 = 75 (Correct!)

Our values satisfy all three equations! That means we've successfully solved the system. We can now confidently say that (25, 40, 55) is the solution to our system of equations. Isn't it satisfying when all the pieces come together? Now that we've walked through this problem step-by-step, let's think about what we've learned and how we can apply these techniques to other problems.

Tips and Tricks for Solving Systems of Equations

Solving systems of equations can feel like a puzzle, but with the right strategies, it becomes much more manageable. Here are some tips and tricks to help you along the way:

  • Stay Organized: Write neatly and keep your equations aligned. This helps prevent errors and makes it easier to review your work.
  • Label Your Equations: Labeling your equations (e.g., Equation 1, Equation 2, Equation 3) makes it easier to refer to them in your steps.
  • Choose the Easiest Variable to Eliminate: Look for variables that have coefficients that are easy to work with or that are already opposites. This can simplify the elimination process.
  • Multiply Carefully: When multiplying an equation, make sure to multiply every term on both sides of the equation.
  • Check Your Work: Always check your solution by plugging the values back into the original equations. This is the best way to catch mistakes.
  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with solving systems of equations. Try different types of problems to challenge yourself.

These tips can make a big difference in your ability to solve systems of equations accurately and efficiently. Remember, solving these problems is like building a skill; each problem you solve makes you a little bit better. So, keep practicing, and you'll become a pro in no time!

Conclusion

So, there you have it! We've successfully solved a system of three equations with three unknowns using the elimination method. We walked through each step, from understanding the problem to verifying our solution. Remember, the key to mastering systems of equations is practice and a systematic approach. By following the steps we've outlined and using the tips we've shared, you'll be well-equipped to tackle any system of equations that comes your way. Whether you're solving math problems for school or applying these skills to real-world situations, the ability to solve systems of equations is a valuable asset.

We hope this guide has been helpful and has made the process of solving systems of equations a little less daunting and a lot more fun. Keep practicing, keep exploring, and most importantly, keep enjoying the world of mathematics! Until next time, happy solving!