Solve S=v_{0}t+at^2/2

$$S = v _ { 0 } t + \frac { a t ^ { 2 } } { 2 }$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{2\left(tv_{0}-S\right)}{t^{2}}\text{, }&t\neq 0\\a\in \mathrm{R}\text{, }&S=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$v_{0}t+\frac{at^{2}}{2}=S$$

Subtract v_{0}t from both sides.

$$\frac{at^{2}}{2}=S-v_{0}t$$

Multiply both sides of the equation by 2.

$$at^{2}=2S-2v_{0}t$$

The equation is in standard form.

$$t^{2}a=2S-2tv_{0}$$

Divide both sides by t^{2}.

$$\frac{t^{2}a}{t^{2}}=\frac{2S-2tv_{0}}{t^{2}}$$

Dividing by t^{2} undoes the multiplication by t^{2}.

$$a=\frac{2S-2tv_{0}}{t^{2}}$$

Divide 2S-2v_{0}t by t^{2}.

$$a=\frac{2\left(S-tv_{0}\right)}{t^{2}}$$

Solve for S

$$S=\frac{t\left(at+2v_{0}\right)}{2}$$