System of two equations solver
Solve ax + by = c and dx + ey = f, with the steps (Cramer's rule).
How it works
Determinant Δ = ae − bd. If Δ ≠ 0: x = (ce − bf) / Δ and y = (af − cd) / Δ. If Δ = 0 there are no solutions or infinitely many.
Results are for guidance only and do not replace professional advice.
Worked example
With the default values:
- a (1st equation)2
- b3
- c13
- d (2nd equation)1
- e-1
- f-1
you get:
- Solutionx = 2 · y = 3
- Determinant Δ = ae − bd-5
- Δx = ce − bf-10
- Δy = af − cd-15
Frequently asked questions
How does the “System of two equations solver” tool work?
Determinant Δ = ae − bd. If Δ ≠ 0: x = (ce − bf) / Δ and y = (af − cd) / Δ. If Δ = 0 there are no solutions or infinitely many.
What do I need to enter?
Just fill in: a, b, c, d, e, f. The result updates as you type.
Can you give an example?
a (1st equation): 2; b: 3; c: 13; d (2nd equation): 1; e: -1 → Solution: x = 2 · y = 3; Determinant Δ = ae − bd: -5.
Is this calculator free, and is my data private?
Yes. HOD Calc is completely free with no sign-up. The calculation runs in your browser: nothing you type is sent or stored.