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
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.
- 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.
- 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?) ?
- 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.