The equation of any line in 2, except a line parallel to the y-axis, can be written in the form
where m is the gradient or slope of the line, and c is its y-intercept; that is, (0, c) is the point at which the line crosses the y-axis. (See the first sketch below.)
In the particular case that the line cuts the y-axis at the origin, its equation has the simple form
as c = 0 in this case. (See the second sketch below.)
Another special case occurs when m = 0. Then the line is parallel to the x-axis, and its equation is of the form
where c is the y-intercept. (See the third sketch below.)
Finally, the equation of a line parallel to the y-axis cannot be written in the form y = mx + c, but it can be written as
where (a, 0) is the point at which the line crosses the x-axis. (See the final sketch below.)
In both cases (1.1) and (1.2) above, the equation of a line in the plane can be written in the form
for some real numbers a, b and c, where a and b are not both zero.
Thus any line in 2 has an equation of the form (1.3); conversely, any equation of the form (1.3) represents a line in 2.
Equation of a line
The general equation of a line in 2 is
ax + by = c,
where a, b and c are real, and a and b are not both zero.
Determine the equation of the line with gradient −3 that passes through the point (2, −1).
Using the formula for the equation of a line when given its gradient and one point on it, we find that the equation of this line is
We can rearrange this in the form
For each of the following pairs of points, determine the equation of the line through them.
(a) (1, 1) and (3, 5).
(b) (0, 0) and (0, 8).
(c) (0, 0) and (4, 2).
(d) (4, −1) and (2, −1).
(a) Since (1, 1) and (3, 5) lie on the line, its gradient is
Then, since the point (1, 1) lies on the line, its equation must be y − 1 = 2(x − 1), or y = 2x − 1.
(b) Both these points have x-coordinate 0, so they lie on the line with equation x = 0, the y-axis.
(c) Since the origin lies on the line, its equation must be of the form y = mx, where m is its gradient.
Since (4, 2) lies on the line, its coordinates must satisfy the equation of the line. Thus 2 = 4 m, so
Hence the equation of this line is
(d) Both these points have y-coordinate −1, so they lie on the line with equation y = −1.