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.
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Input
Enter the coefficients of two linear equations to solve the system with Cramer''s rule and find the solution (x, y).
First equation
Second equation
Use the form a1x + b1y = c1 and a2x + b2y = c2.
Result
Unique solution
x = 3 , y = 2
Determinant D
-5
Dx
-15
Dy
-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.
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