Systems of Linear Equations - Graphical Approach

The html 5 applet below helps you explore and interpret graphically the solutions of a 2 by 2 systems of linear equations of the form

a1x + b1y = c1
a2x + b2y = c2.

Sliders are used to change the values of the coefficients a1, b1, c1, a2, b2 and c2 in order to explore different systems of equations. A graphical interpretation is used here in order to give a complete picture to 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 HTML 5 Applet

a1 = -10+10

b1 = -10+10

c1 = -10+10

a2 = -10+10

b2 = -10+10

c2 = -10+10

Click on "draw" to start.

The starting (default values) of the coefficients are: a1 = 1 , b1 = 3, c1 = 5 and a2 = 2, b2 = -2, c2 = 2 which gives the system of equations:

x + 3y = 5
2x - 2 y = 2

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

It is easy to check that the point of intersection (2 , 1) of the two lines is solution to the system of equations made up of the equations of the lines.
  1. Use the sliders to set the constants a1, b1, c1 ,a2, b2 and c2 to different values so that the two lines have other point of intersection (x , y ). Check that the ordered pair (x , y), is the solution to the system (find a solution by any algebraic method such as substitution, elimination, cramer's rule,...) and compare.


  2. Set a1, a2, b1 and b2 to values such that a1*b2 - a2*b1 = 0 (example: a1 = 2, b1 = 4, a2 = 1 and b2 = 2). 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 (example: a1 = 1, b1 = 1, c1 = 1 and a2 = 2, b2 = 2 and c2 = 2. How many solutions the system has ?Explain.


More references to systems of linear equations.