Gradient of Line

In general, why is it assumed there is only one line passing through two points with a given slope, rather than an infinite amount of overlapping lines?

A line in space is obtained by placing a ruler in space at some fixed point with a fixed direction, and drawing the line it gives. That is, every point of a line can be described in the form

x(t)=x0+tv𝑥(𝑡)=𝑥0+𝑡𝑣

where x0𝑥0 is the starting point in space, and t𝑡 is the "time it takes to get to x𝑥 in the direction of the velocity vector v𝑣."

If a line given parametrically by

x(t)=x0+tv𝑥(𝑡)=𝑥0+𝑡𝑣

passes through P𝑃 and Q,𝑄, we may without loss of generality assume it passes through P𝑃 at time 0 and Q𝑄 at time 1--we may scale and/or shift t as necessary to accomplish this. So we have

x(t)=x0+tv𝑥(𝑡)=𝑥0+𝑡𝑣

x(0)=P𝑥(0)=𝑃

x(1)=Q𝑥(1)=𝑄

from which we conclude

x0=P𝑥0=𝑃

v=Q−P.𝑣=𝑄−𝑃.

That is, x(t)𝑥(𝑡) must given by

x(t)=P+t(Q−P),𝑥(𝑡)=𝑃+𝑡(𝑄−𝑃),

and there is a unique line between P𝑃 and Q.

Based on the well-established mathematical principle, we propose the following understanding:

A straight line defined by two specific points can include infinitely many other points. This is because only one straight line can pass through any two fixed points.

For instance, consider a straight line named Line X that passes through two given points, A and B. If you choose any other two points on this line, say C and D, and remove the original points A, B, and Line X, you can still redraw the same straight line through points A and B using the new points C and D. By repeating this process, it becomes evident that a straight line contains infinitely many points. This concept is supported by mathematical proofs available on platforms like Math Stack Exchange.