System of Linear Equations in Two Variables

Drag the points and/or the lines so that the lines you prefer are in the coordinate system. The line may also by entered as equations by typing their parameters for either the slope-intercept form or the standard form. You may mix entry methods as you please, for instance entering one line by cicking-and-dragging and the other as an equation, or by using different equation forms, or you may enter both lines using the same entry method.

Exploration 1:
Find the point of intersection of these two lines: line A, passing through the points (0, –3) and (6, 6), and line B, passing through the points (2, 4) and (–4, 7).

1. Add 1 to the x-values of both of the given points of line A. By what amount does the x-value of the point of intersection change?
2. Starting with the original points, add 1 to the x-values of both of the given points of line B. By what amount does the point of intersection change?
3. Starting with the original points, add 1 to the x-values of the given points of both lines. By what amount does the point of intersection change?
4. Suppose you want the y-value of the point of intersection to increase by 1. Do you need to add 1 to the y-values of the original points of line A, or those of line B, or both?

Exploration 2:
Graph the lines given by the equations 3x – 5y = 10 and y = 0.6x – 2.

1. What geometric concept is displayed by the graphs of the equations?
2. Change the y-intercept of one of the lines so that the lines no longer coincide. Are the lines still parallel?
3. Now change the slope of the other line so that the lines intersect at some point. Is it possible to make the lines intersect at a common y-intercept?
4. Enter the following parameters for the standard forms of the lines: for line A, 1, 0, and 3, and for line B, 0, 1, and 5. How can these lines be described?