Two equations, two unknowns
A pair of linear equations in x and y usually meets at a single point -- the solution. This calculator solves the 2x2 system by Cramers rule, returns x and y, and tells you when the lines are parallel (no solution) or the same line (infinitely many).
Cramers rule
Write the system as a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The determinant is D = a₁b₂ - a₂b₁. When D is not zero there is one solution: x = (c₁b₂ - c₂b₁) / D and y = (a₁c₂ - a₂c₁) / D.
When there is no single answer
If D = 0 the lines are parallel or identical. Same slope and different intercept means no solution; same slope and same intercept means the lines coincide and every point on the line solves both equations.
Related algebra tools
For the slope and intercept of one line through two points, use the slope calculator. For a single quadratic, the quadratic formula calculator.
Worked example
For x + y = 5 and 2x - y = 1, the determinant is (1)(-1) - (2)(1) = -3, giving x = 2 and y = 3 -- the point where the two lines cross.
FAQ
What if my equations are not in standard form?
Rearrange each into a x + b y = c first, moving all variable terms to the left and the constant to the right, then read off a, b and c.
Can it handle three unknowns?
This solves the 2x2 case. A third variable needs a 3x3 method, which this tool does not cover.