Solving 3x3 Systems: A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving into the world of solving 3x3 systems of equations. Don't worry, it might sound intimidating, but trust me, it's totally manageable! We'll break down the process step by step, making it easy for you to grasp. We'll tackle the given system, find the values of x, y, and z, and make sure you feel confident with this type of problem. So, let's get started, guys!

Understanding the Basics: What is a 3x3 System?

So, what exactly is a 3x3 system? Well, it's simply a set of three linear equations, each with three variables (x, y, and z in our case). Our goal is to find the unique values of x, y, and z that satisfy all three equations simultaneously. Think of it like finding the single point in 3D space where three planes intersect. Each equation represents a plane, and the solution is the point where all three planes meet. In our specific problem, we have the following system:

  • x + 2y - z = -3
  • 2x - y + z = 5
  • x - y + z = 4

These equations, when graphed, represent three planes in 3D space. The solution to the system is the point (x, y, z) where these three planes intersect. The intersection point, if it exists, gives us the unique values for x, y, and z that satisfy all three equations simultaneously. If the planes don't intersect at a single point (e.g., they are parallel or intersect in a line), then the system has either no solution or infinitely many solutions. For our specific problem, we will find a unique solution.

Method of Elimination: Our Weapon of Choice

There are several methods to solve a 3x3 system, like substitution or using matrices. However, we will use the elimination method in this example because it's often the most straightforward. The elimination method involves strategically adding or subtracting the equations to eliminate one variable at a time. The goal is to reduce the system to simpler equations until we can isolate each variable. With a bit of practice, you'll find it becomes quite intuitive.

Let's start by labeling our equations:

  • Equation 1: x + 2y - z = -3
  • Equation 2: 2x - y + z = 5
  • Equation 3: x - y + z = 4

Notice that we already have a good situation with the z variable! In equations 2 and 3, we have +z, and with equation 1 we have -z, so we can eliminate z pretty easily. Let's get started, shall we?

Eliminating z: The First Step

Our first target is to eliminate one of the variables. Looking at the equations, we can eliminate z easily by adding Equation 2 and Equation 3. This is because the z terms have opposite signs. Let's do it:

Equation 2: 2x - y + z = 5 Equation 3: x - y + z = 4

Adding these equations, we get:

3x - 2y + 2z = 9 (Let's call this Equation 4)

We can also eliminate z by adding Equation 1 and Equation 2:

Equation 1: x + 2y - z = -3 Equation 2: 2x - y + z = 5

Adding these equations, we get:

3x + y = 2 (Let's call this Equation 5)

Now we have two new equations (Equations 4 and 5) with only x and y. Our system is getting simpler!

Eliminating z: The Second Step

Now we will eliminate z using Equation 1 and Equation 2.

Equation 1: x + 2y - z = -3 Equation 2: 2x - y + z = 5

Adding Equation 1 and Equation 2, we get:

3x + y = 2 (This is our Equation 5)

This is very important; now we have two equations with two variables. This is great progress, guys!

Solving for x and y

Now we have a simpler 2x2 system:

  • Equation 4: 3x - 2y = 9
  • Equation 5: 3x + y = 2

To eliminate x, we can multiply Equation 5 by -1 and then add it to Equation 4. Let's do it:

Multiply Equation 5 by -1: -3x - y = -2

Now, add this modified equation to Equation 4:

  • Equation 4: 3x - 2y = 9
  • -3x - y = -2

Adding these gives us:

-3y = 7

Solving for y, we get:

y = -7/3

Now that we have y, let's substitute it into Equation 5 to solve for x:

3x + y = 2 3x + (-7/3) = 2 3x = 2 + 7/3 3x = 13/3 x = 13/9

Fantastic! We've got x and y! Just one more step to go!

Solving for z

Now, we've got the values for x and y, and we're just one step away from the final solution. Substitute the values of x and y we found back into any of the original equations containing z. Let's use Equation 3:

Equation 3: x - y + z = 4

Substitute x = 13/9 and y = -7/3:

13/9 - (-7/3) + z = 4 13/9 + 7/3 + z = 4 13/9 + 21/9 + z = 4 34/9 + z = 4 z = 4 - 34/9 z = 36/9 - 34/9 z = 2/9

The Solution: The Grand Finale

And there you have it! We've found the values of x, y, and z that satisfy the system of equations:

  • x = 13/9
  • y = -7/3
  • z = 2/9

This means the three planes defined by the original equations intersect at the point (13/9, -7/3, 2/9). You did it! High five! You have successfully solved a 3x3 system of equations using the elimination method. Remember, practice makes perfect. The more problems you solve, the more comfortable you'll become with these systems.

Tips for Success: Keep These in Mind

  • Organize your work: Keep track of your equations and the steps you're taking. This helps prevent errors and makes it easier to find mistakes. Write down each step clearly. This will make your job easier. This includes labeling each equation and the new equations you produce. This will also allow you to go back and check your work. This is very useful. When you get the correct answer, you can just check your work and make sure each step makes sense. If you made a mistake, you can easily find where you messed up. This is better than starting from scratch. It will save a lot of time, believe me.
  • Double-check your arithmetic: Small errors in calculations can lead to incorrect answers. Always double-check your math, especially when adding, subtracting, multiplying, or dividing.
  • Be patient: Solving 3x3 systems can take a few steps. Don't get discouraged if it takes a while to find the solution. The process is systematic, and you'll get there if you're patient and persistent.
  • Practice, practice, practice: The more problems you solve, the more comfortable you'll become with the elimination method. Work through various examples to build your skills and confidence.
  • Understand the concept: While the elimination method is effective, make sure you understand what you are doing and why it works. Knowing the underlying principles will help you adapt the method to different types of systems and problems.

Keep these tips in mind, and you'll be well on your way to mastering 3x3 systems! Keep practicing, and you'll see how much easier this gets. If you have any questions, don't hesitate to ask.

Final Thoughts

Great job, everyone! Solving a 3x3 system might seem like a mountain to climb, but with these steps, it's actually a series of hills. You've taken on a challenging problem and come out on top. Just remember to break down the problem into manageable steps, use the elimination method with precision, and always double-check your work. The more you practice, the more you'll get the hang of it. Keep up the amazing work, and enjoy the satisfaction of conquering those equations!