The Linear command solves system of linear equations.

Optional arguments can contain expressions which must not be zero, notz, and the method for solving the system.

If method is not specified, Linear tries to dispatch to an algorithm according to the type of the system. The possible methods correspond to the following types of equations (the check is performed in this order).

If the method is not specified and the system is not of one of the above types, Linear uses the default universal method, which is a primitive fraction-free algorithm.

Most algorithms are intended to be used on large sparse systems, however, they also perform well on dense systems. The exception is the RationalDense method that is specifically optimized for dense systems of rational numbers, especially overconstrained systems with infinitely many solutions.

The algorithms used are Gaussian elimination with pivoting for stability for the numeric coefficients and primitive fraction-free for the algebraic and radical coefficients.

Many of the methods can also be called directly as exports of the SolveTools:-LinearSolvers module.

$\mathrm{with}\left(\mathrm{SolveTools}\right)\:$

$\mathrm{Linear}\left(\left\{x+y\,x-y-2\right\}\,\left\{x\,y\right\}\right)$

$\left\{x=1\,y=\mathrm{-1}\right\}$

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

$\mathrm{Linear}\left(\left\{x+y-5.\,4x-3y-2\right\}\,\left\{x\,y\right\}\right)$

$\left\{x=2.428571428\,y=2.571428571\right\}$

$\mathrm{Linear}\left(\left\{x+y-5\,4x-3y-2-\mathrm{I}\right\}\,\left\{x\,y\right\}\,\mathrm{method}=\mathrm{ComplexFloat}\right)$

$\left\{x=2.428571428+0.1428571429\mathrm{I}\,y=2.571428571-0.1428571428\mathrm{I}\right\}$

$\mathrm{Linear}\left(\left\{x+y-5\,4x-3y-2\right\}\,\left\{x\right\}\right)$

$\mathrm{Linear}\left(\left\{x+y+z-5^{\frac{1}{2}}\,4x-3y-2\right\}\,\left\{x\,y\,z\right\}\right)$

$\left\{x=\frac{1}{2}+\frac{3y}{4}\,y=y\,z=-\frac{1}{2}-\frac{7y}{4}+\sqrt{5}\right\}$

$\mathrm{Linear}\left(\left\{x+2\mathrm{RootOf}\left(v^2-w\,v\right)wy\,5xw-3y+7\right\}\,\left\{x\,y\right\}\right)$

$\left\{x=-\frac{14w\left(10w^3-3\mathrm{RootOf}\left({\mathrm{\_Z}}^2-w\right)\right)}{100w^5-9}\,y=\frac{7\left(10\mathrm{RootOf}\left({\mathrm{\_Z}}^2-w\right)w^2-3\right)}{100w^5-9}\right\}$

The method option was updated in Maple 2023.