Linear Algebra Powertool

Plotting Systems of Linear Systems in 2 & 3 Variables

Worksheet by Russell Blyth

Systems of two linear equations in two variables can have one of three solution types

> with(plots);

Warning, the name changecoords has been redefined

Two intersecting lines

> implicitplot({x+y=3,x-y=-1},x=-1..3,y=-1..3);

> solve({x+y=3,x-y=-1},{x,y});

Two coinciding lines

> implicitplot({x+y=3,2*x+2*y=6},x=-3..3,y=-3..3);

> solve({x+y=3,2*x+2*y=6},{x,y});

Two parallel lines

> implicitplot({x+y=3,x+y=1},x=-3..3,y=-3..3);

> solve({x+y=3,x+y=1},{x,y});

Two intersecting planes having a line as solution set

> implicitplot3d({x-3*z=-3,2*x-z=-2},x=-3..3,y=-3..3,z=-3..3,axes=boxed,orientation=[49,64]);

> solve({x-3*z=-3,2*x-z=-2},{x,y,z});

Three intersecting planes meeting in a single point

> implicitplot3d({x-3*z=-3,2*x-z=-2,x+2*y=1},x=-3..3,y=-3..3,z=-3..3,axes=boxed,orientation=[17,74]);

> solve({x-3*z=-3,2*x-z=-2,x+2*y=1},{x,y,z});

Three planes intersecting in a line is a second possibility (a third is that all three planes coincide)

> implicitplot3d({x-3*z=-3,2*x+2*y-z=-2,x+2*y+2*z=1},x=-3..3,y=-3..3,z=-3..3,axes=boxed,orientation=[-41,74]);

> solve({x-3*z=-3,2*x+2*y-z=-2,x+2*y+2*z=1},{x,y,z});

If two of the planes are parallel and do not coincide, then there are no solutions. But there are other ways to have no solutions:

Three planes, not parallel, intersecting pairwise in parallel lines

> implicitplot3d({x-3*z=-3,2*x-5*y-z=-2,x+2*y-5*z=1},x=-3..3,y=-3..3,z=-3..3,axes=boxed,orientation=[17,74]);

> solve({x-3*z=-3,2*x-5*y-z=-2,x+2*y-5*z=1},{x,y,z});

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