How this calculator works
Substitution isolates a variable in one equation and replaces it in the other. If the first x coefficient is zero, solve its y equation first. Scaled row reduction supplies the numerical answer and detects contradiction or dependence. Very nearly dependent systems are treated as singular at a relative coefficient tolerance of 1e-12.
Formula
x = (c - by)/a, then d(c - by)/a + ey = f, when a is non-zero.
Worked example
From x + 2y = 8, isolate x = 8 - 2y. Put this in 2x - y = 1: 16 - 5y = 1, so y = 3 and x = 2.
Advertisement
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.