### Systems of Linear Equations

Graphs the two lines associated with a system of two linear equations in two variables, showing the solution of the system as the point where the lines intersect (if a unique solution exists).
How to use   ||   Examples   ||   Other Notes

How to use
• Enter coefficients into the corresponding text input fields (enter the value for coefficient A into the text input field marked "A=", for B into the field marked "B=", etc.).
• Click the "Graph" button (this button can also be used to refresh the graph)
• To reset the graph and all input fields to default values, click the "Clear" button
The text input fields can accept any decimal number input.

The buttons under the graph allow various manipulations of the graph coordinates.

Examples
 Independent:    x+2y=8 3x-2y=0 Dependent:    2x-y=0 -4x+2y=0 Inconsistent:    2x-y=0 2x-y=2

Other Notes
The labels under the text input field tell whether the system is independent, dependent, or inconsistent, and in the independent case also give the coordinates of the solution point (where the two lines cross). This information is determined as follows:
• If AE-DB=0, then the system is either dependent or inconsistent. To determine these cases:
• If CE-FB=0, then the system is dependent. In this case, the two equations describe the same line, and all points on that line are solutions of the system. The label will read: "solutions: entire line".
• Otherwise, the system is inconsistent. In this case, the two equations describe parallel lines, which never cross, so there is no solution point. The label will read: "no solution".
• Otherwise, the system is independent, and the two equations describe nonparallel lines, which cross exactly once. The point (x,y) where the two lines cross is the solution point for the system. The coordinates of the point (x,y) are determined using Cramer's Rule: x=(CE-FB)/(AE-DB) and y=(AF-DC)/(AE-DB). The label will display this solution point.