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Spacetime is a mathematical model in which the three dimensions of space are combined with a fourth dimension of time into a single continuum or a vector space called Minkowski space. Doing so enables us to treat space and time in the same way, and thus greatly simplifies a number of theories in physics, especially the theory of relativity.
In classical physics, it was believed that time was a universal constant and thus independent of motion through space. Special relativity demonstrated that time actually varies with motion, since as we approach the speed of light, our time slows down relative to a stationary observer. General relativity demonstrated that gravitational fields can have a similar impact upon time. Both theories can be more simply expressed with the notion of the change in the rate of time as being due to its bending into space. Without treating space and time as part of the same continuum, such a formulation is impossible.
In relativistic mechanics, the vector space conception of spacetime is hugely helpful in understanding certain predictions of the theory, such as time dilation. Velocity in space and the passage of time collectively form the gradient with respect time of the vector space, and everything in the universe is moving at the same four-dimensional velocity (the speed of light) within that space. Time dilation simply means that more velocity in the three space dimensions requires a correspondingly reduced velocity in the time dimension.