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+tvπ₯(π‘)=π₯0+π‘π£
where x0π₯0 is the starting point in space, and tπ‘ is the "time it takes to get to xπ₯ in the direction of the velocity vector vπ£."
If a line given parametrically by
x(t)=x0+tvπ₯(π‘)=π₯0+π‘π£
passes through Pπ and Q,π, we may without loss of generality assume it passes through Pπ at time 0 and Qπ at time 1--we may scale and/or shift t as necessary to accomplish this. So we have
x(t)=x0+tvπ₯(π‘)=π₯0+π‘π£
x(0)=Pπ₯(0)=π
x(1)=Qπ₯(1)=π
from which we conclude
x0=Pπ₯0=π
v=QβP.π£=πβπ.
That is, x(t)π₯(π‘) must given by
x(t)=P+t(QβP),π₯(π‘)=π+π‘(πβπ),
and there is a unique line between Pπ 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.
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