In the equation x = vt, v is a constant. Watch this:
Acceleration is constant:
(1.) a(t) = a_0
Integrating, we have velocity:
(2.) v(t) = a_0 * t + v_0
v_0 is the initial velocity. Integrating again, we have position:
(3.) s(t) = (a_0/2) * t^2 + v_0 * t + s_0
s_0 is the initial position. Note, if we let a_0 = 0, s_0 = 0, (3.) becomes this familiar expression:
(4.) s(t) = v_0 * t
This naturally implies that v(t) = v_0.
One needs to distinguish between constants, various types of functions, and dependent and independent variables.