Is Linear Equation Solver free?

Yes. Linear Equation Solver on MathsUK is completely free, runs in the browser with no sign-up and no installation.

← Back to all tools

Linear Equation Solver

ax + b = c — a guided solution showing every rearrangement.

2x + 3 = 11

Solution:

x = 4

Solution steps

Rearrange: 2x = 11 − 3 = 8
Divide by the coefficient: x = (11 − 3) / 2
Calculate: x = 8 / 2 = 4
Check by substitution: 2x = 8 + 3 = 11 ✓

Examples

What is a linear equation?

A linear equation in one variable is an equation of the form ax + b = c, where a, b and c are given numbers and x is the unknown. The word "linear" comes from the Latin linea — a line — because the graph of the expression ax + b is a straight line. The goal is to find the value of x that makes both sides equal.

The idea is simple: isolate x. First move b to the right-hand side by subtracting it, then divide both sides by a. Whatever you do to one side you must do to the other — otherwise the equality breaks. When a term crosses the equals sign, its sign flips: a "+5" that moves across becomes "−5".

Everyday examples

In a classroom: a class has 30 pupils, 5 are absent. How many are present? The equation is x + 5 = 30, so x = 25.
Shopping: a T-shirt costs £x, you paid £100 and got £28 change. The equation: x + 28 = 100, so x = 72.
Speed: a car travelled for 4 hours at speed x and covered 240 km. The equation: 4x = 240, so x = 60 km/h.

Common mistakes

Forgetting to flip the sign when a term crosses the equals sign! If b = +5 on the left-hand side, on the right-hand side it becomes −5.
Dividing only one side by a. Every operation must be applied to both sides.
Ignoring the a = 0 case.If the coefficient of x is 0, it isn't really a linear equation any more — there are either infinitely many solutions or none at all.
Getting the sign wrong with a negative coefficient. Dividing by a negative number flips the sign of the result.

💼 Uses: in parking (working out waiting times), in money calculations (change, price before VAT), in general algebra, in physics (speed, time, distance), and in countless school exercises. Linear equations are the first tool taught in algebra, because they are the foundation for everything that comes after.

How to use it

Enter the coefficients a, b, c of the equation ax + b = c.
The tool moves b to the right-hand side (flipping its sign) and divides by a.
When a = 0: if b = c there are infinitely many solutions, otherwise there is no solution.
The solution is shown rounded to 4 decimal places.

Related tools

✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications