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The linear system defined by eqns is solved for the unknowns vars. If a solution exists, the solution is returned as a list of equations. If the system is underdetermined, the solver will parametrize the solutions in terms of one or more of the unknowns.

Currently, the solver uses a number of special algorithms in addition to ordinary Gaussian elimination. All algorithms are geared to large sparse systems though they also perform well on dense systems. The special algorithms (selected automatically) are for:

(a) rational coefficients, using a "primitive" fraction-free algorithm;

(b) floating-point coefficients, using Gaussian elimination with partial pivoting for stability;

(c) rational function coefficients over the rationals, more generally, an algebraic number field, using a "primitive" fraction-free algorithm.

$\mathrm{solve}\left(\left\{ax+by=3\,x-y=b\right\}\,\left[x\,y\right]\right)$

$\left[\left[x=\frac{b^2+3}{a+b}\,y=-\frac{ab-3}{a+b}\right]\right]$

$\mathrm{solve}\left(\left\{\mathrm{I}x+y=1\,\mathrm{I}x-y=2\right\}\,\left[x\,y\right]\right)$

$\left[\left[x=-\frac{3\mathrm{I}}{2}\,y=-\frac{1}{2}\right]\right]$