Equation of a line
Definition
The equation of a line is an equation which connects the x and the y values for every point on the line. When graphed, all the pairs (x,y) that are a solution of this equation form the line.
Using the gradient formula, the position of a general point (x,y) on a line with gradient m passing through , is given by .
Rearranging, we find the equation of the line is
If we further rearrange this equation, we can get the gradient-intercept form of the equation of the line:
In general form, the equation of a line is where a, b, d are constants.
Observations
A horizontal line has gradient . This means the equation of a horizontal line is (if we know that it goes through point ) or in general .
A vertical line has an equation (if we know that it goes through point ) or in general . These lines do not have a gradient.
In the next applet, you can move A and B to modify the line and observe the changes in the equations. You can switch from the Point-Gradient form and the Gradient-Intercept form using the checkboxes.
The y-intercept
In the applet above, make the Gradient-Intercept form equation visible and modify points A and B. Watch what happens with the equation as you do it, and try to find a relationship between the equation and the intersection of the line with the y-axis. Explain your observations.