Enter coefficients for a linear system in the form ax + by + cz = d.
What this 3 system of equations calculator does
This calculator solves a system of three linear equations with three unknowns: x, y, and z. It is useful for algebra, linear algebra homework, engineering problems, economics models, and any situation where multiple linear constraints must be solved together.
Unlike basic tools that only return one number, this page checks whether your system has:
- A unique solution (one exact intersection point),
- No solution (inconsistent equations), or
- Infinitely many solutions (dependent equations).
How to enter equations correctly
Use standard form
Each equation must be entered as: ax + by + cz = d. Put the coefficient of each variable in the matching input box. If a variable is missing, enter 0 for its coefficient.
- Example: 4x + 0y - 7z = 10 means
a = 4,b = 0,c = -7,d = 10. - Decimals are supported, such as 1.5, -0.2, and 3.14159.
- Negative numbers are supported directly with a minus sign.
Worked example
The preloaded sample system is:
- 2x + y - z = 8
- -3x - y + 2z = -11
- -2x + y + 2z = -3
Click Calculate and the solver returns the unique solution: x = 2, y = 3, z = -1.
How the calculator solves the system
Gaussian elimination
Internally, the tool converts your equations into an augmented matrix and uses Gaussian elimination with pivoting. This method reduces the matrix to an upper-triangular form and then performs back-substitution to compute variable values.
Rank-based checks for edge cases
When the determinant is effectively zero, the calculator does not stop there. It checks matrix rank to determine whether equations are contradictory (no solution) or dependent (infinitely many solutions).
Why this matters for learning
Many students memorize elimination steps but are not sure what the final result means. Seeing clear output categories helps connect the algebra to geometry:
- Unique solution: three planes intersect at one point.
- No solution: planes do not all intersect together.
- Infinite solutions: at least one equation is redundant, creating a line or plane of valid points.
Common mistakes to avoid
- Mixing up constants and coefficients (placing d under x, y, or z).
- Forgetting to enter zero for a missing variable term.
- Using extremely rounded decimals that change system behavior.
- Accidentally typing a space or leaving a field empty.
FAQ
Can I use fractions?
Yes. Convert fractions to decimals before entering them (for example, 1/3 becomes 0.333333).
Does this work for nonlinear equations?
No. This tool is specifically for linear systems in three variables.
What if I get “infinite solutions”?
That means the equations are not fully independent. You can express one variable in terms of another (or others), producing many valid triples (x, y, z).
What if I get “no solution”?
Your equations conflict. At least one equation contradicts the others, so there is no single point that satisfies all three simultaneously.