x+y+z=1 and x+y=1 and x+y-z=1 are 3 equations. Algebraically, I get that there are infinite solutions.

Graphically, on desmos 3D Visualisation - Equations - Solutions | Desmos

I get this. The lines x=k and y=1-k are on the plane z=0 So, I see that the solutions are many points moving in a line along the plane z= 0.

Would a line joining these points with different k not be a solution to the simultaneous equation? Instead of saying it is infinite, could we not say it is a line?

Or is it that I should treat each k separately, then I get a point for k=0, another point for (0, 1, 0)

k = 1 (1, 0, 0) etc and many other real numbers in between and these infinite points are the infinite distinct solutions?