The first few applets show the linear systems:

a x + b y = e
c x + d y = f

The red line represents the first equation and the blue line represents the second.

The purple dot represents the solution to the system, if there is (only) one.

The green dotted line represents the following linear combination of the two equations: p(a x + b y) + q(c x + d y) = p e + q f

There are two variants of the applet. This one Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) has sliders representing a, b, c, d, e, f, p and q. This allows you to experiment with gradually changing one variable and seeing what happens, but has the disadvantage that each variable must be in the range -10 to 10.

This one: Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) works the same but without sliders; double-click on the variables on the left and edit them. (Make sure you leave everything to the left of, and including, the equals sign.)