If you want to derive that equation of motion in a serious way you always use calculus.

You start from the definition:

$a(t)= \frac {dv(t)}{dt}$

You integrate both members between 0 and $t_f$

$\int_{0}^{t_f} {a(t)dt} = \int_{0}^{t_f} {v'(t)dt}$

and you finally get:

$v(t_f) = v(0)+\int_{0}^{t_f} {a(t)dt}$

If $a(t)=ct$ then

$v(t_f) = v(0)+ at_f$

Algebra is used in high schools where students do not know yet calculus but the derivation of a motion equation using algebra is considered more a trick than serious math.

If you want to derive that equation of motion in a serious way you always use calculus.

You start from the definition:

$a(t)= \frac {dv(t)}{dt}$

You integrate both sides from 0 to $t_f$

$\int_{0}^{t_f} {a(t)dt} = \int_{0}^{t_f} {v'(t)dt}$

and you finally get:

$v(t_f) = v(0)+\int_{0}^{t_f} {a(t)dt}$

If $a(t)=ct$ then

$v(t_f) = v(0)+ at_f$

Algebra is used in high schools where students do not know yet calculus but the derivation of a motion equation using algebra is considered more a trick than serious math.