Simultaneous Equations (2 Variables) Calculator

Work out simultaneous equations (2 variables) instantly with clear inputs, formula shown and shareable results.

About Simultaneous Equations (2 Variables) Calculator

Cramer's rule solves a two-by-two system directly: x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D where D = a₁b₂ − a₂b₁. A zero determinant means the lines are parallel.

Formula

Cramer's rule

D = a₁b₂ − a₂b₁, x = (c₁b₂ − c₂b₁)/D, y = (a₁c₂ − a₂c₁)/D

Worked example

With the default values below, Simultaneous Equations (2 Variables) Calculator returns x of 3.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

First equation coefficients a, b, c (for ax + by = c): 2, 3, 12
Second equation coefficients a, b, c: 4, -1, 10

Results

x: 3.00000000
y: 2.00000000
Determinant a₁b₂ − a₂b₁: -14.00000000
First equation left-hand side: 12.00000000

Step by step

1Values used
2Cramer's rule
3x= 3.00000000
4y= 2.00000000
5Determinant a₁b₂ − a₂b₁= -14.00000000
6First equation left-hand side= 12.00000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Solve 2x + 3y = 12 and 4x − y = 10.

D = −14, giving x = 3 and y = 2.

What does a zero determinant mean?

The two lines have the same gradient: either they never meet (no solution) or they coincide (infinitely many).