Solving Systems in 3 Variables: A Complete Walkthrough

Solving systems of equations with three variables can seem daunting, but with a systematic approach, it becomes manageable. Here's a detailed walkthrough:

Understanding 3-Variable Systems 🧐

A system of three equations with three variables (typically x, y, and z) represents three planes in 3D space. The solution to the system is the point where all three planes intersect. There are a few possible scenarios:

• Unique Solution: The planes intersect at a single point.
• No Solution: The planes do not intersect at any common point (parallel planes, etc.).
• Infinite Solutions: The planes intersect along a line or are the same plane.

Methods for Solving 🛠️

There are two primary methods:

1. Substitution
2. Elimination (also known as addition)

We'll focus on the elimination method, as it's often more efficient for 3x3 systems.

Step-by-Step Elimination Method 🪜

1. Label Your Equations: Number your equations for easy reference.

     (1)  2x + y - z = 8
     (2) -3x - y + 2z = -11
     (3) -x + y + 2z = -3

2. Eliminate a Variable (Round 1): Choose a variable to eliminate. Let's eliminate 'y'. Add equations (1) and (2) to eliminate 'y':

     (1)  2x + y - z = 8
     (2) -3x - y + 2z = -11
     ------------------
     (4) -x + z = -3

   Now, add equations (2) and (3) to eliminate 'y' again:

     (2) -3x - y + 2z = -11
     (3) -x + y + 2z = -3
     ------------------
     (5) -4x + 4z = -14
3. Eliminate the Same Variable (Round 2): Now you have a system of two equations with two variables (x and z):

     (4) -x + z = -3
     (5) -4x + 4z = -14

   Multiply equation (4) by -4:

     (6) 4x - 4z = 12

   Add equations (5) and (6) to eliminate 'z':

     (5) -4x + 4z = -14
     (6)  4x - 4z = 12
     ------------------
     0 = -2
4. Solve for Remaining Variables: Since we got 0 = -2, this system has no solution. The three planes do not intersect at a single point. If we *did* get a valid equation (e.g., 0 = 0 or a solvable equation like x = 2), we would substitute back into the equations to find the values of the other variables.

Example with a Solution 💡

Let's solve a system that *does* have a solution:

  (1)  x + y + z = 6
  (2)  2x - y + z = 3
  (3)  x + 2y - z = 2

1. Eliminate 'z'. Add (1) and (3):

     (4) 2x + 3y = 8

   Add (1) and (2) multiplied by -1:

     (5) -x + 2y = 3

2. Solve the 2x2 system. Multiply (5) by 2:

     (6) -2x + 4y = 6

   Add (4) and (6):

     7y = 14  =>  y = 2

3. Substitute y=2 into (4):

     2x + 3(2) = 8  =>  2x = 2  =>  x = 1

4. Substitute x=1 and y=2 into (1):

     1 + 2 + z = 6  =>  z = 3

Therefore, the solution is x = 1, y = 2, and z = 3, or (1, 2, 3).

Tips for Success ✨

• Stay Organized: Keep your work neat and clearly labeled.
• Double-Check: Verify your calculations at each step.
• Choose Wisely: Select the easiest variable to eliminate.
• Practice: The more you practice, the better you'll become!

By following these steps and practicing regularly, you can confidently solve systems of equations with three variables. Good luck! 🚀