The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
GeoGebra, Markus Hohenwarter and Judith Preiner

Linear Equations

GeoGebra provides multiple representations of mathematical objects and can help your students to discover connections between equations and their graphical representation. Let's investigate the simplest of all examples, a linear equation y = m x + b and its corresponding line in the applet below. There, you see two numbers m and b which are also represented by sliders in the graphics window. We used these parameters to create line g by entering the equation g: y = m x + b in the input field.

Move the sliders to change the parameters m and b and see how the line and its equation adapt dynamically. This way of investigating parameters with sliders can be used for lines, all kind of conic sections or functions like f(x) = b cos(m x) in GeoGebra.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)

Editor's note, May 2014: For an HTML5 version of the above applet, click here.

Let's come back to our simple example of the linear equation and use the construction above to visualize a system of two linear equations.

  1. Create the points A = (-2, 1) and B = (6, 3)by either using the input field or the tool Tool: New Point"New Point".
  2. Now, draw a line through A and B by choosing the tool Tool: Line through two Points "Line through two points" and clicking on both points.
  3. Afterwards you can create the intersection point of the two lines by using the tool Tool: Intersect two Objects "Intersect two objects".

The two equations that you have now created can be seen as a system of linear equations, with their intersection point as the solution. GeoGebra allows you to modify these equations dynamically by activating the tool Tool: Move"Move" and moving the points or the sliders. In this way, you are able to investigate all possible situations including the interesting special cases of no solution and infinitely many solutions.

This approach of visualizing equations can be applied to functions as well. To show the equation x2 = 4 you could enter the functions f(x) = x^2 and g(x) = 4 and intersect them. Another way would be to take f(x) = x2 − 4 and intersect this function with the x-axis, i.e. find the roots of this function. Furthermore, your functions can also include slider parameters or non-polynomial expressions like sin(x), abs(x), exp(x) or log(x).