D is not 0
There is a unique solution. The two lines intersect at one point, and that point has x and y coordinates.
Math
2x2 system calculator
Enter two linear equations and get the solution with Cramer's rule, determinants, interpretation and verification.
System of equations
Enter each equation's coefficients in the form ax + by = c.
Normalized system
a1x + b1y = c1
a2x + b2y = c2
Examples
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How to interpret the result
The main determinant indicates whether the lines intersect, are parallel or represent the same line.
D = 0 with different constants
There is no solution. The lines have the same slope but pass through different points.
D = Dx = Dy = 0
There are infinitely many solutions. Both equations describe the same line, or one equation adds no restrictions.
Frequently asked questions
What is a 2x2 system?
It is a system with two linear equations and two unknowns, usually x and y. The solution is the pair of values that satisfies both equations.
What does a zero determinant mean?
If the main determinant D is zero, the lines do not intersect at a unique point. The system may have no solution or infinitely many solutions.
What is Cramer's rule?
It is a method that uses determinants. For a 2x2 system, D, Dx and Dy are calculated; if D is not zero, x = Dx / D and y = Dy / D.
Can there be infinitely many solutions?
Yes. This occurs when both equations represent the same line or one equation adds no restriction because it is 0 = 0.
How is the solution verified?
Substitute the calculated x and y values into both equations. If both equalities hold, the result is consistent.
What happens if an equation is 0 = 5?
That equation is impossible. Because no pair of values can satisfy it, the entire system has no solution.