How can the first order kinematic (SUVAT) equations be derived?

We start with the following two observations about an object undergoing constant acceleration. First, its acceleration is equal to the change in its velocity over time, hence,

a=(v-u)/t.

Rearranging gives the first SUVAT equation,

v=u+at.

Secondly, we observe that the average velocity of the object is equal to the distance it travels over time. The average velocity of an object undergoing constant acceleration is the average of its initial and final velocities, hence,

(u+v)/2=s/t.

Substituting the value of v in the first SUVAT equation, we have,

(2u+at)/2=s/t.

Rearranging, we have the second SUVAT equation,

s=ut+(at^2)/2.

To derive the third equation, the original equations are rearranged to give,

v-u=at

and

v+u=2s/t.

These equations can be multiplied to give,

(v+u)(v-u)=2as.

Multiplying out the brackets and rearranging gives the third SUVAT equation,

v^2=u^2+2as.