Simultaneous Equations

In its most familiar form, a linear system is expressed as a set of simultaneous linear equations. A typical example is

and a solution is a pair (x,y) which makes each equation true. Such a pair is said to simultaneously solve each equation.

A linear system can involve any number of equations and any number of variables. In the example above, with two variables, we can visualize the system in terms of the plane of points (x,y). Now each equation defines a line in the plane, consisting of all the points for which the equation is true. To be a solution to the system of equations, a point must lie on the lines for both equations.

In the activity below, you can explore these ideas by defining your own pair of equations, and then testing whether individual points in the plane satisfy either equation. You define the equations by typing the coefficients into red and blue colored boxes on the screen, and you select test points by clicking in the graph with the mouse. You can also have the graph of each equation appear on the screen by clicking the appropriate button.

Here are a couple variations on using the interaction above.

1. Enter new equations, but do not click the buttons to create the graphs for these equations. Then see whether, by clicking and testing points, you can discover solutions to each equation, as well as a simultaneous solution to both equations.

2. Parallel Lines. Do you know how to identify parallel lines? If the coefficients of x and y are the same for both equations but the numbers on the opposite side of the equations are different, the resulting lines will be parallel. For example, here are a pair of parallel lines:

Can you find a solution to the system of equations in this case?

3. Coincident Lines. Sometimes two equations describe the same line. For example, this happens when one equation is a multiple of the other, like this:

What can you say about solutions to the system of equations in this case?