You have seen that the geometric object called a line is uniquely defined by two points.
In a coordinate system, the points acquire an x-value and a y-value, the coordinates of the points. Every point on the line has a pair of coordinates, not just the points that are displayed.
On this line it is easy to see that every y-value is exactly three larger than the x-value of the same point.
Instead of labeling every point to show the relationship between the x- and y- values, the line itself can be labeled with an equation that expresses the relationship. This is the equation of the line.
Every line has an equation. These equations are all written in a form known as the slope-intercept form.
There are many forms of equations for lines. Some of the important ones that you will be using are the slope-intercept form, the point-slope form, and the standard form. Most lines can have their equations written in each of these forms.
Some special cases have special forms. The equations for horizontal lines and vertical lines have simpler forms than other lines.
This simulation will allow you to experiment with the different forms of the equations of lines. It loads with a basic line, two points on the line, and the standard form of the equation of the line with the parameters already filled in.
Experiment by dragging the points to different locations …
… or by dragging the line itself.
When you drag a point, the line will pivot about the other point.
When you drag the line, the line moves as a whole and doesn’t change its orientation, or slope.
As you drag the points or the line around, the coefficients in the selected equation will update to reflect the line in its current position …
… as will the coordinates of the points on the line.
You can also make adjustments by typing numbers directly into the input-output fields. Click the field you want to change, type a whole number or a decimal number, and press Enter.
The line is redrawn to reflect the changes you make after you press Enter.
You can choose the form in which to display the equation of the line by selecting the desired form from the list.
Experiment with all the fields. Try changing just one of the parameters at a time, then changing all of the parameters of a single equation at once. See if you can predict the effect of a change before you make it. Above all, have fun and learn! 1