ALGEBRA

Solve a system of two linear equations in two unknowns. Enter the coefficients for both equations.

a1 (x in eq 1)

b1 (y in eq 1)

c1 (eq 1 equals)

a2 (x in eq 2)

b2 (y in eq 2)

c2 (eq 2 equals)

Solution (x, y)

—

Where the lines cross.

Usage Tip

Two lines either cross once (one solution), run parallel (none), or coincide (infinitely many) — the determinant tells which.

THE MATH

a1·x + b1·y = c1 and a2·x + b2·y = c2
Solved by Cramer rule using the determinant a1·b2 − a2·b1

A system of two linear equations represents two lines; the solution is where they intersect.
If the determinant is zero, the lines are parallel or identical and there is no single solution.

Enter the equations in the form ax + by = c.
A non-zero determinant gives one unique solution.
Zero determinant means no solution or infinitely many.

Practical Notes

A system of two linear equations has two unknowns, usually x and y, and the solution is the point where the two lines cross. Each equation is entered in the form ax + by = c. The calculator finds the single point that satisfies both equations at once.

There are three possible outcomes. Most systems have exactly one solution. If the lines are parallel and never meet, there is no solution, and if both equations describe the same line, there are infinitely many. These cases show up when the determinant of the coefficients is zero. You can check any answer by substituting the x and y values back into both original equations.

Spotted an error or have a suggestion for this calculator? Let us know →