Question

How do you find the equation of a line with Guassian elimination given two points?

Answer 1

The solution is similar to the 2-point form of the equation for a line:

y = (y2 -y1)/(x2 -x1)·x + (y1) -(x1)(y2 -y1)/(x2 -x1)

Explanation:

Using the two points, write two equations in the unknowns of the equation of the line.

For example, you can use the equation ...

y = mx + b

Then for the points (x1, y1) and (x2, y2) you have two equations in m and b:

b + (x1)m = (y1)

b + (x2)m = (y2)

The corresponding augmented matrix for this system is ...

`\left[\begin{array}c1&x1&y1\\1&x2&y2\end{array}\right]`

____

The "b" variable can be eliminated by subtracting the first equation from the second. This puts a 0 in row 2 column 1 of the matrix, per Gaussian Elimination.

0 + (x2 -x1)m = (y2 -y1)

Dividing by the value in row 2 column 2 gives you the value of m:

m = (y2 -y1)/(x2 -x1)

This value can be substituted into either equation to find the value of b.

b = (y1) -(x1)(y2 -y1)/(x2 -x1) . . . . . substituting for m in the first equation