System of equations
Solve a system of two linear equations, with the intersection shown on a graph.
Enter both equations' coefficients – the solution and graph appear instantly.
How to solve a system of two equations
A system of two linear equations in two unknowns has the form ax + by = c and dx + ey = f. Each equation is a line in the plane, and the solution of the system is their intersection point. There are several methods – substitution, elimination, or using determinants (Cramer's rule):
x = (c·e − b·f) / (a·e − b·d) • y = (a·f − c·d) / (a·e − b·d)
The denominator a·e − b·d is the determinant of the system. When it's not zero, there is exactly one solution – the lines intersect at a single point.
When there is no solution, or infinitely many
- one solution determinant ≠ 0 → the lines intersect at one point
- no solution the lines are parallel and distinct (they never meet)
- infinitely many both equations describe the same line (they coincide)
The calculator computes the determinant and distinguishes all three cases with a graph.
Tip: the solution (x, y) is the point where both lines cross on the graph.