This applet visualizes a system of three simultanous equations in the variables x,y and z:

a11 x + a12 y + a13 z = b1

a21 x + a22 y + a23 z = b2

a31 x + a32 y + a33 z = b3

The intersection points of each plane with the three x-,y- and z-axis are given by:

plane1 (green): x = b1/a11 (y=z=0), y = b1/a12 (x=z=0), z = b1/a13 (x=y=0)

plane2 ( red ): x = b2/a21 (y=z=0), y = b2/a22 (x=z=0), z = b2/a23 (x=y=0)

plane3 (blue ): x = b3/a31 (y=z=0), y = b3/a32 (x=z=0), z = b3/a33 (x=y=0)

The dark green line represents the intersection of planes 1 and 3 and the red line the intersection
of planes 2 and 3. The fat golden point represents the solution of the system of three linear
equations. You can freely manipulate the points of each plane with the three x-,y- and z-axis. (Actually
in the case depicted here the third plane is parallel to the y axis, so in this case a32=0.)

Created with Cinderella