On the previous page, we gave several standard methods for finding lines parallel or perpendicular to other lines, lines that go through two points, etc.  All these examples were limited to two dimensions.  A more general method of finding a line that goes through two points is the Determinant Method, which is presented here.

| x    y    1|
| x₁   y₁   1| = 0
| x₂   y₂   1|

Expand the determinant by whatever method you learned and you have the equation of the line through the points

(x₁, y₁) and
(x₂, y₂).

Example: Find the equation of the line through
(2,3) and (-4,1).

|  x    y    1|
|  2    3    1| = 0
| -4    1    1|

Expanding by minors across the top row:

x·|3  1| - y·| 2 1| + 1·| 2  3| = 0
  |1  1|     |-4 1|     |-4  1|

x·(3·1-1·1) - y·(2·1-1·-4) + 1·(2·1-3·-4) = 0
x(3-1) - y(2+4) + (2+12) = 0
2x - 6y + 14 = 0

Divide every term by 2:

x - 3y + 7 = 0

The webmaster and author of the Math Help site is Graeme McRae.
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