System of Equations Calculator (2 Variables, Cramer's Rule)
By Santaji GadepatilinUpdated November 6, 2026
A system of two linear equations ax + by = c has one solution where the two lines cross, none if they are parallel and infinitely many if they are the same line. This system of equations calculator solves for x and y with Cramer's rule and shows the determinant, which is zero exactly when there is no unique solution.
How to use the system of equations calculator
Write both equations in the form a·x + b·y = c and enter the six coefficients. The calculator returns the determinant and the solution. If the determinant is 0, the equations have no unique solution.
Cramer’s rule for two equations
D = a₁b₂ − a₂b₁ · x = (c₁b₂ − c₂b₁) / D · y = (a₁c₂ − a₂c₁) / D
Worked example
Solve 2x + 3y = 13 and x − y = −1:
- D = 2 × (−1) − 1 × 3 = -5
- x = (13 × (−1) − (−1) × 3) ÷ (-5) = 2
- y = (2 × (−1) − 1 × 13) ÷ (-5) = 3
- Check: 2(2) + 3(3) = 13 ✓ and 2 − 3 = −1 ✓
Solving by elimination
- Multiply the equations so one variable has equal and opposite coefficients.
- Add the equations to eliminate that variable and solve for the other.
- Substitute back to find the first variable.
Types of solution
| Determinant | Lines | Solutions |
|---|---|---|
| D ≠ 0 | Intersect at one point | Exactly one |
| D = 0, equations proportional | Same line | Infinitely many |
| D = 0, not proportional | Parallel | None |
Frequently asked questions
How do you solve two equations with two unknowns?
By substitution, elimination or Cramer's rule. With Cramer's rule, x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D, where D = a₁b₂ − a₂b₁.
What does a determinant of zero mean?
The lines are parallel (no solution) or identical (infinitely many solutions). Check whether one equation is a multiple of the other.
How do I enter an equation like y = 2x + 1?
Rearrange it to the form ax + by = c first, i.e. −2x + y = 1, so a = −2, b = 1 and c = 1.
How can I check the answer?
Substitute x and y back into both original equations — each left-hand side should equal its right-hand side.