Systems of Linear Equations - Graphical Approach

This large window java applet helps you explore the solutions of 2 by 2 systems of linear equations of the form

a1x + b1y = c1
a2x + b2y = c2.

A graphical approach is taken here in order to give a complete picture on solving systems of equations with the existing algebraic methods (elimination, cramer's rule,...). Also included in this site, a tutorial on solving systems of linear using analytical methods. Interactive Tutorial Using Java Applet

What is the relationship between the geometric positions of the lines representing the two equations of a system and the solutions of the system ?

  1. Use the scrollbar to set the constants a1, a2, b1, b2, c1 and c2 so that the two lines have a point of intersection (x , y ). Check that the ordered pair (x , y), is the solution to the system (find a solution by any method you wish : substitution, elimination, cramer's rule,...) and compare.


  2. Set a1, a2, b1 and b2 to values such that a1*b2 - a2*b1 = 0. How are the two lines positioned with respect to each other ? How many solutions the system has ? If (the determinant) a1*b2 - a2*b1 = 0, what is the relationship between the slopes of the two lines (are they equal for example?) ?


  3. Set all 6 constants to values such that a1/a2 = b1/b2 = c1/c2. How many solutions the system has ?Explain.


More references to systems of linear equations.