How this calculator works
Elimination combines equations to cancel a variable. Multiplying the second equation by a and the first by d eliminates x. Row reduction with scaled pivots handles zero coefficients and distinguishes inconsistent rows from dependent rows. Very nearly dependent systems are treated as numerically singular at a relative coefficient tolerance of 1e-12.
Formula
(ae - db)y = af - dc; x = (c - by)/a when a is non-zero.
Worked example
From x + y = 5 and x - y = 1, subtract the second equation from the first: 2y = 4, so y = 2. Substitution back gives x = 3.
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Frequently asked questions
Can there be no solution?
Yes. Parallel inconsistent equations produce a contradiction such as 0 = 1.
When are there infinitely many solutions?
When the independent equations do not constrain both variables and there is no contradiction.
Are rounded steps used to calculate the answer?
No. Display text is rounded separately; the numerical solver retains its floating-point precision.