# Number of solutions

A linear system has exactly one solution, no solution, or infinitely many. During elimination, a row 0 = 5 means none, and a row of zeros means free variables.

Maths (university) · Part 08: Vectors, matrices and linear systems · Card 9 of 9 · Last updated 2026-10-19

## Analogy

Two lines can cross once, be parallel, or lie exactly on top of each other.

## Example

**Two bad systems**

x + y = 2 and 2x + 2y = 4:
same line, infinitely many solutions.
x + y = 2 and x + y = 3:
parallel, no solution.

## In practice

Tells you whether a model is consistent before you try to solve it.

## Practice

### Flashcards

- **How many solutions can a linear system have?** One, none, or infinitely many.

### Interview questions

**Explain the three possible outcomes for the number of solutions to a linear system with an example of each.**

- One solution: lines cross once
- None: parallel lines, a row 0 = non-zero
- Infinitely many: same line, a free variable
- Elimination reveals which case occurs

## Questions people ask

### How can I tell there is no solution?

During elimination you reach a row of zeros on the left with a non-zero number on the right, such as 0 = 1. That statement is impossible, so no choice of unknowns can satisfy all the equations. The system is called inconsistent.

### What is a free variable?

When elimination leaves fewer useful equations than unknowns, some unknowns can take any value, and the others depend on them. For x plus y equals 2, y is free, and x equals 2 minus y. That gives infinitely many solutions.

## Related concepts

- [Linear system](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-6)
- [Augmented matrix](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-7)
- [Gaussian elimination](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-8)
- [Solving with the inverse](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-4)

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