---
title: "Systems of Linear Equations and Their Solution Sets"
description: "A linear system has no solution, exactly one, or infinitely many, and nothing else. Reading which case you are in, and describing an infinite solution set, is the starting point for the whole subject."
canonical: https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te
source: https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te.md
retrieved: 2026-09-12
---

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# Systems of Linear Equations and Their Solution Sets

A linear system has no solution, exactly one, or infinitely many, and nothing else. Reading which case you are in, and describing an infinite solution set, is the starting point for the whole subject.

Subject: Mathematics · Area: Linear Algebra · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te

## Ready when they can

- Classify a two-variable system by the geometry of its lines
- Describe an infinite solution set with a free parameter
- Give an inconsistent system and say what makes it inconsistent

## Lesson: One point, no point, or a whole line

You already know a solution to two linear equations is where graphs cross, one point. A system gathers equations and asks which points make every one true at once. To test a pair, try it in each equation: 2 and negative 1 make 3 x minus y equals 7 and x minus 2 y equals 4 true. A pair failing even one equation is not a solution.

Two lines leave three options: one crossing is one solution, parallel gives none, and a shared line gives infinitely many. x plus y equals 1 beside x plus y equals 2 cannot both hold. x plus y equals 4 beside 2 x plus 2 y equals 8 share every point.

**Example.** Solve x plus y equals 7 and x minus y equals 3. Add to kill y, giving 2 x equals 10, so x equals 5. Put 5 in the first equation, giving y equals 2. Check: 5 plus 2 is 7 and 5 minus 2 is 3. For a shared line, name it with a free parameter like y equals 4 minus x.

Elimination adds or subtracts whole equations to kill one variable. A result of 0 equals 5 means parallel lines and no solution, while 0 equals 0 means one shared line. Two equations in three unknowns leave one variable free, so the answer is none or infinitely many, never exactly one.

**Recap.** One crossing is one solution, parallel lines give none, and a shared line gives infinitely many.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp)
- [Simultaneous Equations](https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz)

## Opens up

- [Dimension and the Rank-Nullity Theorem](https://lightmysky.com/learn/mathematics/dimension-and-the-rank-nullity-theorem-mt_aMG8PcwPAR)
- [Gaussian Elimination and Row Echelon Form](https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm)