Then x has disappeared from the equation. If the remaining statement is true, every value of x works; if it is false, none does. This says which rather than dividing by zero.

Math & Statisticsalgebra

Linear Equation Solver

A linear equation of the form ax plus b equals c, solved with the substitution shown as a check. The two degenerate cases, no solution and infinitely many, are named rather than hidden behind a division by zero.

Also called: solve for x, linear equation.

x = 5. Substituting back: 3 times 5 plus 4 gives 19, which is 19 as required.

Substituted back

Rearranged

x = (19 - 4) / 3

Nature of the equation

exactly one solution

How this is calculated

Method and background

How this is calculated

Move the constant across and divide. The only interesting part is what happens when a is zero: if b also equals c the equation is true for every x, and if it does not the equation is true for none. Both are reported explicitly, because a solver that returns infinity or an error in those cases has thrown away the actual answer.

x = (c - b) / a

a: Coefficient of x
b: Constant on the left
c: Right-hand side

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a standard solve

a (coefficient of x): 3
b (constant on the left): 4
c (right-hand side): 19

arithmetic identity with substitution check

Open this example

x cancels and the statement holds

a (coefficient of x): 0
b (constant on the left): 5
c (right-hand side): 5

boundary

Open this example

x cancels and the statement fails

a (coefficient of x): 0
b (constant on the left): 5
c (right-hand side): 7

degenerate case

Open this example

Method and limits

What it assumes

Real coefficients.

What it deliberately does not model

One equation in one unknown. Simultaneous equations are a different tool.

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Frequently asked questions

What if a is zero?: Then x has disappeared from the equation. If the remaining statement is true, every value of x works; if it is false, none does. This says which rather than dividing by zero.