You will be using the equations of motion all the time. Surely you want to see their derivation/proof?
The v-t equation basically comes directly from the definition of acceleration.
For displacement, we can start from , where vave refers to the average velocity during the duration t. Since for uniform acceleration motion,
Another way to derive equation (2) is by considering the trapezium area under the v-t graph.
From equation (2), we just need to get rid of v by substituting into equation (2).
To derive the v-s equation, we just need to get rid of t by substituting into equation (2).