---
title: "What Is a Linear Equation?"
url: https://www.learningloftinstitute.com/glossary/linear-equation
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Glossary › What Is a Linear Equation?"
---
# What Is a Linear Equation?
> **Answer.** A linear equation is one in which every variable appears only to the power one and no two variables are multiplied together. 3x + 2 = 11 is linear; xy = 6 and x² = 9 are not. Plotted on axes, a linear equation in two variables always gives a straight line.
## Key facts
| | |
| --- | --- |
| Definition | An equation in which every variable appears only to the power one, with no variables multiplied together, so its graph is a straight line. |
| Where students meet it | Grade 7 mathematics, when equations first carry a letter on both sides, and again at GCSE and IGCSE as the equation of a straight line. |
| Also known as | First-degree equation |
## The test that decides it
Two conditions, and an equation has to pass both. Every variable must appear to the power one — no squares, no cubes, nothing under a root sign, nothing sitting in a denominator. And no two variables may be multiplied by one another.
Run the awkward cases through it. x² = 9 fails on the first condition and there is nothing to argue about. xy = 6 is the interesting one: both x and y are to the power one, so it passes the first condition, and it still fails, because the two variables are multiplied together.
5/x = 10 fails as well, though it looks harmless. Rewriting it as 5x⁻¹ = 10 makes the reason visible — the variable is at the power −1, not 1.
- 3x + 2 = 11 — linear
- 2x + 5y = 20 — linear, two variables
- x² = 9 — not linear, the variable is squared
- xy = 6 — not linear, two variables multiplied
- 5/x = 10 — not linear, x sits at the power −1
## Why the word is about a picture
The name is borrowed from the graph rather than the algebra. Every pair of values satisfying 2x + 5y = 20 lies on one straight line; the pairs satisfying y = x² lie on a curve. That is the whole reason for the word, and it is why those two conditions are the ones that matter — break either and the graph bends.
In one variable the picture is simpler still, and it comes with a guarantee: a linear equation such as 3x + 2 = 11 has exactly one solution. There are two collapses worth recognising. 2x + 1 = 2x + 5 has no solution, because the variable vanishes and leaves a false statement, and 2x + 4 = 2(x + 2) is true for every value of x.
## Linear does not mean easy
Students read the word as a comment on difficulty, which holds until it does not. 4(3x − 7)/5 + 2 = 3(x + 1) is linear — one variable, power one — and it takes several careful lines to unpick. x² = 9 is not linear and takes one.
The word classifies the equation, not the effort. What it buys you is certainty about method: any linear equation in one variable is solved by collecting terms and dividing. No factorising, no formula, and no second answer to go hunting for.
## Questions
### Is 3x + 2y = 12 a linear equation?
Yes. Both variables are at the power one and neither multiplies the other, so it passes the test. Having two variables means it does not have one solution — it has infinitely many pairs of values, and every one of them lies on the same straight line when plotted.
### Why is x² = 9 not linear?
Because the variable is squared, which breaks the power-one condition. The consequence shows up on the graph, where y = x² is a curve rather than a line, and in the answer, where x² = 9 has two solutions, 3 and −3. Linear equations in one variable have exactly one.
### Is 5/x = 10 a linear equation?
No. Dividing by x is the same as multiplying by x⁻¹, so the variable sits at the power −1. Equations of that shape have curved graphs that run away towards the axes rather than straight lines, and they are usually called reciprocal or rational equations instead.
### What is the difference between a linear equation and a linear expression?
An equation has an equals sign and can be solved; an expression cannot. 3x + 2 is a linear expression, and there is nothing to find — you can only simplify or substitute. 3x + 2 = 11 is a linear equation, and x = 3. Exam papers use the two words precisely.
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