Enter the coefficients of each equation in the form ax + by = c (or ax + by + cz = d). Decimals and fractions like 3/4 are fine, and use 0 for a missing term. Answers are exact, so you get ⅓ rather than 0.333…
Simultaneous equations are satisfied by the same values at once. For two equations, each is a straight line, and the solution is where they cross. Parallel lines never cross (no solution); identical lines overlap everywhere (infinitely many solutions).
Cramer’s rule uses determinants. For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the determinant D = a₁b₂ − a₂b₁. Then x = Dx ÷ D and y = Dy ÷ D, where Dx = c₁b₂ − c₂b₁ and Dy = a₁c₂ − a₂c₁. If D = 0 there’s no single answer.
Example: 2x + 3y = 13 and x − y = −1. D = 2(−1) − 1(3) = −5, Dx = 13(−1) − (−1)(3) = −10, Dy = 2(−1) − 1(13) = −15, so x = 2 and y = 3. Check: 2·2 + 3·3 = 13 ✓.
Three equations work the same way with 3 × 3 determinants. The solver uses exact fractions (Gaussian elimination) to find the answer, then shows the determinants as working.
By substitution (rearrange one equation for x and substitute it into the other), elimination (add or subtract multiples of the equations to cancel a variable), or Cramer’s rule with determinants. They all give the same answer; this calculator shows the determinant working.
Enter 0 as its coefficient. For example, y = 5 in a two-variable system is 0x + 1y = 5.
Rearrange it so all the variables are on the left: −2x + y = 1. Then enter a = −2, b = 1, c = 1.
The equations contradict each other, such as x + y = 2 and x + y = 5. For two equations the lines are parallel and never meet.
At least one equation is just a multiple or combination of the others, such as x + y = 2 and 2x + 2y = 4, so there are not enough independent equations to pin down a single answer.
