Linear System Solver (3 Variables)
A linear system solver with three variables is a mathematical tool that finds the values of x, y, and z that satisfy three linear equations at the same time. These systems are fundamental in algebra, engineering, physics, economics, and computer science.
Solve a 3-Variable Linear System
Enter coefficients for:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
What Is a Linear System with Three Variables?
A linear system with three variables consists of three equations where each equation includes the same three unknowns: x, y, and z.
Each equation represents a plane in three-dimensional space. The solution to the system is the point where all three planes intersect.
General Form
The standard form is:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Possible Types of Solutions
- One unique solution – planes intersect at one point
- No solution – planes do not intersect at one point
- Infinitely many solutions – planes overlap
How the Calculator Works
Cramer's Rule (Mathematical Explanation)
Cramer's Rule uses determinants to solve linear systems. If the determinant of the coefficient matrix is not zero, the system has a unique solution.
x = Dx / D
y = Dy / D
z = Dz / D
Worked Example
Solve:
x + y + z = 6
2x − y + z = 3
x + 2y − z = 3
Solution:
x = 1, y = 2, z = 3
Geometric Interpretation
Each equation is a plane. A unique solution exists when all planes intersect at exactly one point.
Real-World Applications
- 3D physics simulations
- Engineering systems
- Chemical reaction balancing
- Economic equilibrium models
- Computer graphics
Common Mistakes
- Forgetting one variable
- Incorrect determinant calculation
- Assuming a solution always exists
Frequently Asked Questions
Can coefficients be decimals?
Yes.
What if the determinant is zero?
The system has no unique solution.
Is this exact?
Results are shown in accurate decimal form.
Conclusion
The Linear System Solver (3 Variables) is an essential algebra tool for solving complex systems quickly and accurately.
Understanding how it works builds strong foundations for advanced mathematics and real-world problem solving.