Solving 3x3 Linear Systems
What is it? 🤔
A 3x3 linear system has three equations with three variables (like x, y, z). We solve it to find where three planes intersect in 3D space. This is crucial for real-world problems with three constraints, like business models, physics, and engineering.
How to Solve: Elimination Method
- Pair and Eliminate: Choose two pairs of equations. Eliminate the same variable from both pairs.
- Create a 2x2 System: You'll get two new equations with just two variables. Solve this smaller system.
- Back Substitute: Plug your two found values into one original equation to find the third variable.
- Check Your Solution: Verify your (x, y, z) triple works in all three original equations.
Example 1
System:
(1) x + y - z = 0
(2) 2x - y + z = 3
(3) -x + 2y + 2z = 6
Step 1: Eliminate 'y'. Add (1)+(2): 3x = 3 → x = 1.
Multiply (1) by 2 and add to (3): x + 4z = 6.
Step 2: With x=1, 1 + 4z = 6 → z = 5/4.
Step 3: Substitute into (1): 1 + y - 5/4 = 0 → y = 1/4.
Solution: (1, 1/4, 5/4)
Example 2
System:
(1) 2x + y = 1
(2) y - z = 2
(3) x + z = 3
Step 1: Notice (2) and (3) are ready. Add them: x + y = 5.
Step 2: Use with (1): x + y = 5 and 2x + y = 1. Subtract: -x = 4 → x = -4.
Step 3: From (1): 2(-4) + y = 1 → y = 9. From (2): 9 - z = 2 → z = 7.
Solution: (-4, 9, 7)
Common Mistakes ⚠️
- Inconsistent Elimination: Not eliminating the same variable from both equation pairs.
- Sign Errors: Forgetting negative signs when multiplying equations.
- Arithmetic Slips: Simple addition/subtraction errors.
- Not Checking: Always verify your solution in all three original equations!
Tips & Tricks
- Look for Zeros: Choose a variable to eliminate that already has a coefficient of 1 or -1.
- Stay Organized: Label your equations (1), (2), (3) and write each step clearly.
- Check Early: Verify your 2x2 system solution before back-substituting.
How to Practice
Start with systems that have "nice" integer solutions. Practice the elimination process until it's comfortable. Then move to problems with fractions. Create your own systems by choosing an (x, y, z) solution and making up three equations. Try online generators for unlimited practice!