Quick start
- Write both equations in the form ax + by = c.
- Enter a₁, b₁, c₁ for the first equation and a₂, b₂, c₂ for the second.
- Read the main determinant D, then Dx and Dy, and finally x and y.
Cramer's rule for two equations
D = a₁b₂ − a₂b₁. Replace the x column with the constants to get Dx = c₁b₂ − c₂b₁, and replace the y column to get Dy = a₁c₂ − a₂c₁. Then x = Dx ÷ D and y = Dy ÷ D.
Example
For 2x + 3y = 12 and x − y = 1: D = 2·(−1) − 1·3 = −5, Dx = 12·(−1) − 1·3 = −15, and Dy = 2·1 − 1·12 = −10. So x = 3 and y = 2. Check: 2·3 + 3·2 = 12 and 3 − 2 = 1.
When D is zero
A zero determinant means the lines have the same slope. If they are parallel there is no solution; if they are the same line there are infinitely many. Either way there is no single meeting point.
Common questions
Can I use substitution or elimination instead?
Yes, all three methods give the same answer. Determinants are the quickest to compute and check.
What does the solution mean on a graph?
It is the point where the two lines cross.
My equation is y = 2x + 1. How do I enter it?
Rearrange it to −2x + y = 1, so a = −2, b = 1 and c = 1.