Two-Variable Linear System Solver

Solve a system of two linear equations a1x+b1y=c1 and a2x+b2y=c2 with Cramer's rule. Get the solution (x, y), classify it as unique, none, or infinite from the determinant, and see the two lines plotted.

Input

Enter the coefficients of two linear equations to solve the system with Cramer''s rule and find the solution (x, y).

First equation

a1 (x coefficient)

b1 (y coefficient)

c1 (constant)

Second equation

a2 (x coefficient)

b2 (y coefficient)

c2 (constant)

Use the form a1x + b1y = c1 and a2x + b2y = c2.

Result

Unique solution

x = 3 , y = 2

Determinant D

-5

-15

-10

Cramer''s rule: D = a1b2 minus a2b1, x = Dx divided by D, y = Dy divided by D. When D is zero there is either no solution or infinitely many.

How it works

The coefficient determinant is D = a1 times b2 minus a2 times b1.
Replacing each column with the constants gives Dx = c1 times b2 minus c2 times b1 and Dy = a1 times c2 minus a2 times c1.
When D is not zero there is a unique solution: x = Dx divided by D and y = Dy divided by D.
When D is zero and both Dx and Dy are zero, the two lines coincide and there are infinitely many solutions.
When D is zero but Dx or Dy is nonzero, the lines are parallel and there is no solution.

Reviews

Tell us what you think of this calculator.