3 systems of equations calculator

What this 3 systems of equations calculator does

This calculator solves a system of three linear equations with three unknowns: x, y, and z. It accepts decimal values, negative numbers, and integers. You can use it for algebra homework, engineering models, chemistry balancing setups, and basic economics equations where three relationships must be satisfied at once.

In short, if your system looks like this:

• a₁x + b₁y + c₁z = d₁
• a₂x + b₂y + c₂z = d₂
• a₃x + b₃y + c₃z = d₃

the tool determines whether there is exactly one solution, no solution, or infinitely many solutions.

How the solver works

Gaussian elimination under the hood

The calculator uses Gaussian elimination with row operations to reduce the augmented matrix. This method is reliable and widely taught in linear algebra. It also handles edge cases better than simple substitution when coefficients are messy or include decimals.

Possible outcomes

• Unique solution: One exact point where all three equations intersect.
• No solution: The equations are inconsistent (parallel/conflicting planes).
• Infinitely many solutions: At least one equation is dependent on the others.

How to use this calculator

1. Enter the coefficient values for each equation row.
2. Put constants on the right side in d₁, d₂, d₃.
3. Click Solve System.
4. Read the result and matrix summary.

If you want to test quickly, click Load Example to prefill a standard system.

Example system

The default values represent:

• 2x + y − z = 8
• −3x − y + 2z = −11
• −2x + y + 2z = −3

The calculator returns: x = 2, y = 3, z = -1.

Tips for accurate input

• Use decimals with a period (example: 3.75).
• Keep variable terms on the left and constants on the right before entering values.
• Double-check signs (+/−). Most mistakes come from sign errors.
• If the determinant is near zero, your system may be singular or nearly singular.

When would you need a 3x3 equation solver?

Common cases include:

• Mixture and concentration problems
• Intersection of three planes in 3D geometry
• Simple equilibrium and budget models
• Electric circuit loop equations
• Data fitting with three unknown parameters

This online 3 variable linear equation solver gives quick answers without requiring a graphing calculator or manual row reduction every time.