MathIsimple

Three-Variable Linear Systems

Master systematic elimination for three-variable systems! Learn to solve systems with x, y, and z, understand geometric representations, and apply structured problem-solving approaches.

10th Grade
Algebra
~60 min
🎮 Interactive Activity: Three-Variable Solver

Solve the system with three variables!

System:

x + y + z = 6
2x - y + z = 3
x + 2y - z = 5
🎮 Interactive Activity: Elimination Steps

Learn the systematic approach to solving three-variable systems!

Step 1: Eliminate one variable

Choose two equations and eliminate one variable (e.g., eliminate z from equations 1 and 2)

Step 2: Eliminate same variable again

Step 3: Solve 2x2 system

Step 4: Back-substitute

1. Introduction to Three-Variable Systems

What Are Three-Variable Systems?

A three-variable linear system consists of three equations with three unknowns (typically x, y, and z). To find a unique solution, you need three independent equations.

General form:

  • Equation 1: a₁x + b₁y + c₁z = d₁
  • Equation 2: a₂x + b₂y + c₂z = d₂
  • Equation 3: a₃x + b₃y + c₃z = d₃
Example 1: A Simple System

Solve: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 5

Step 1: Use elimination to reduce to two variables

Step 2: Solve the resulting 2-variable system

Step 3: Back-substitute to find the third variable

Solution: x = 2, y = 1, z = 3

2. Elimination Method for Three Variables
3. Systematic Solution Strategy
4. Geometric Interpretation
5. Real-World Applications
6. Special Cases
7. Problem-Solving Strategies
8. Augmented Matrix Method
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
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Correct
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1
How many equations are needed to solve a system with three variables?
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2
In the system: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 5, what is the value of x?
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3
Which method is most systematic for solving three-variable systems?
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4
When eliminating a variable, how many times do you need to eliminate it?
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5
If x + y + z = 10, x = 3, and y = 4, what is z?
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6
What is the geometric representation of a three-variable linear equation?
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7
How many planes intersect at a single point for a unique solution?
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8
If the system x + y + z = 6, 2x + 2y + 2z = 12, x + y + z = 10 is solved, what is the result?
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9
After eliminating one variable, what type of system remains?
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10
Which step comes after solving the 2-variable system?
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