Here's the first part
v = u + at
Note that a = change in velocity / change in time
So...... change in velocity / change in time = (v - u ) / t
So we have that
v = u + at
v = u + [ ( v - u) / t ] * t
v = u + (v - u)
v = u + v - u
v = v
Second part
The total displacement, s, is the green area
Notice that the total shaded area of the bottom rectangle is u* t
And the area of the top rectange is (v - u)* t
But only half of this rectangle is shaded...so....the shaded area in the top rectangle is (1/2)(v - u) * t
So.....the displacement is
u * t + (1/2) ( v - u) *t =
u * t + (1/2)v*t - (1/2)u*t =
1/2 v* t + 1/2 u*t =
(1/2) ( u + v) * t = s