Finding y-intercept of linear equation with two points.

Question

If we were given two points on a linear equation $(x_1,y_1),(x_2,y_2)$, it is quite easy to find the slope and use substitution to find the slope intercept form $y=mx+b$, to graph it.

Is it possible to solve for $b$ strictly in terms of $x_1,y_1,x_2$, and $y_2$?

Answer 1

Yes it is. As you say, we can find the slope: $$m=\frac{y_2-y_1}{x_2-x_1}.$$ Thus, $$y_1=mx_1+b=\frac{y_2-y_1}{x_2-x_1}x_1+b,$$ so $$y_1-\frac{y_2-y_1}{x_2-x_1}x_1=\frac{x_2y_1-x_1y_2}{x_2-x_1}=b.$$

Answer 2

Yes, by imposing that the three points$(0,b),(x_1 ,y_1 ),(x_2 ,y_2 )$ are aligned (determinant, or other way you know).