The picture
With two unknowns, each equation is a straight line, and a solution is a point that lies on every line.
- Two lines that cross meet at one point: a unique solution.
- Two parallel lines never meet: no solution.
- Two equations that describe the same line share every point: infinitely many solutions.
With three unknowns the equations are planes, but the same three outcomes are the only possibilities. A linear system can never have exactly two or exactly five solutions.
How elimination reveals the answer
After you eliminate as far as possible, look at the rows of the result:
| You see | Meaning |
|---|---|
| A row like 0 = 3 (no unknowns, non-zero right side) | No solution |
| Every unknown has its own leading term | Unique solution |
| No contradiction, but some unknowns have no leading term | Infinitely many solutions |
Rows that become 0 = 0 are harmless. They just mean one equation was a combination of the others.
Example: no solution
x + y = 1 x + y = 2
Subtract the first from the second and you get 0 = 1, which is impossible. The lines are parallel. Try it in the solver using the “No solution” example.
Example: infinitely many solutions
x + y + z = 3 2x + 2y + 2z = 6
The second equation is just twice the first, so subtracting 2 × Eq 1 leaves 0 = 0. Only one real condition remains on three unknowns. Pick any values for y and z (the free variables) and x follows:
x = 3 − y − z
For example, y = 0, z = 0 gives x = 3, and y = 1, z = 1 gives x = 1. Both satisfy the system.
Rank: the formal test
The rank of a system is the number of independent equations left after elimination, which equals the number of leading terms (pivots). Compare the rank of the coefficients alone with the rank of the coefficients plus the right-hand side:
- If the two ranks differ, there is no solution.
- If they're equal and match the number of unknowns, there is a unique solution.
- If they're equal but smaller than the number of unknowns, there are infinitely many, with (unknowns − rank) free variables.
This is known as the Rouché–Capelli theorem. The solver reports the rank with every result.
Square systems and the determinant
When there are as many equations as unknowns, there's a shortcut: the system has a unique solution exactly when the determinant of the coefficient matrix is not zero. If the determinant is zero, it has either no solution or infinitely many, and elimination tells you which. You can compute determinants with the matrix calculator.
Why it matters in engineering
A “no solution” result usually means conflicting requirements or a modelling error, such as a material balance that can't close. “Infinitely many” means the model is under-specified: you need another measurement or constraint. In structures this is the difference between a statically determinate frame and one that is a mechanism or indeterminate.