System of equations solver

Solve two linear equations in x and y.

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a in ax + by = c

b in ax + by = c

More inputs and assumptions

c in ax + by = c

d in dx + ey = f

e in dx + ey = f

f in dx + ey = f

Answer

…

For two linear equations, determinant = a × e − b × d. Cramer’s rule gives x = (c × e − b × f) ÷ determinant and y = (a × f − c × d) ÷ determinant.

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Rounding changes displayed numbers, not the calculation. Text, fractions, matrices and CSV schedules keep their own formatting. Automatic keeps up to 12 significant digits, with money shown to cents where applicable.

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Formula and assumptions

Supports two equations in x and y. A zero determinant means no unique solution; this tool does not classify every dependent case. Near-zero determinants can make the result sensitive to rounding.

Automatic precision shows numbers to 12 significant digits. Thousands separators such as 1,000 are accepted; use a dot for decimals.

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Updated 2026-09-26

Using System of equations solver

For two linear equations, determinant = a × e − b × d. Cramer’s rule gives x = (c × e − b × f) ÷ determinant and y = (a × f − c × d) ÷ determinant.

Supports two equations in x and y. A zero determinant means no unique solution; this tool does not classify every dependent case. Near-zero determinants can make the result sensitive to rounding.

Next: Quadratic equation solver

A worked example

With a in ax + by = c = 2, b in ax + by = c = 1, c in ax + by = c = 5, d in dx + ey = f = 1, e in dx + ey = f = -1, f in dx + ey = f = 1, the x is 2.

Next: Slope calculator

Before you use the result

Supports two equations in x and y. A zero determinant means no unique solution; this tool does not classify every dependent case. Near-zero determinants can make the result sensitive to rounding.

Example results for a in ax + by = c

These examples use b in ax + by = c: 1; c in ax + by = c: 5; d in dx + ey = f: 1; e in dx + ey = f: -1; f in dx + ey = f: 1. They are reference calculations, not recommended settings.

a in ax + by = c	x
1	3
2	2
4	1.2

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