# Linear system

A set of equations where each unknown appears only to the first power, such as x + y = 5 and x - y = 1. A solution makes every equation true at once.

Maths (university) · Part 08: Vectors, matrices and linear systems · Card 6 of 9 · Last updated 2026-10-19

## Analogy

Two clues about two mystery numbers, which together pin them down.

## Example

**x + y = 5 and x - y = 1**

Add the equations: 2x = 6.
So x = 3.
Then y = 5 - 3 = 2.
Check: 3 - 2 = 1. Correct.

## In practice

Models circuits, budgets, mixtures and balancing problems.

## Practice

### Flashcards

- **Solve x + y = 5 and x - y = 1.** x = 3, y = 2.

### 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

### What does linear mean in a linear system?

Each unknown appears on its own, to the first power, multiplied only by a number. Terms such as x squared, xy, or sine of x make the equation non-linear. Linear equations in two unknowns graph as straight lines, hence the name.

### How do I solve a system by elimination?

Combine equations to remove one unknown. For x plus y equals 5 and x minus y equals 1, adding gives 2x equals 6. Solve that, then substitute back to find the other unknown. Always check your answer in both original equations.

## Related concepts

- [Number of solutions](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-9)
- [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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