Show that x = x1+(x2-x1)t, y = y1 + (y2-y1)t, -∞ < t < ∞ is a parametrization for the line through the points

Question

(x1, y1) and (x2, y2).

b) Find a parametrization for the line segment with endpoints (x1, y1) and (x2, y2)

Answer 1

t=(x-x1)/(x2-x1) and t=(y-y1)/(y2-y1) so...
(x-x1)/(x2-x1)=(y-y1)/(y2-y1)...
(x-x1)(y2-y1)/(x2-x1)=(y-y1)
which is know in point slope form, where (y2-y1)/(x2-x1) =m

(x-x1)m=(y-y1)

Answer 2

Two points define a line. So it is enough to show there are values of t such that the points (x1, y1) and (x2, y2) are on the line.

x = x1 + (x2 - x1)t
y = y1 + (y2 - y1)t
-∞ < t < ∞

for t = 0
x = x1 + (x2 - x1)t = x1 + 0 = x1
y = y1 + (y2 - y1)t = y1 + 0 = y1
So we have the point (x1, y1) on the line.
_______

for t = 1
x = x1 + (x2 - x1)t = x1 + (x2 - x1) = x2
y = y1 + (y2 - y1)t = y1 + (y2 - y1) = y2
So we have the point (x2, y2) on the line.

Since both points are on the line, we have shown that

x = x1 + (x2 - x1)t
y = y1 + (y2 - y1)t
-∞ < t < ∞

is a parameterization for the line thru the given points.