Solve S=ut+1/2gt^2

$$S = u t + \frac { 1 } { 2 } g t ^ { 2 }$$

Solve for g

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

Steps for Solving Linear Equation

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

$$ut+\frac{1}{2}gt^{2}=S$$

Subtract ut from both sides.

$$\frac{1}{2}gt^{2}=S-ut$$

The equation is in standard form.

$$\frac{t^{2}}{2}g=S-tu$$

Divide both sides by \frac{1}{2}t^{2}.

$$\frac{2\times \frac{t^{2}}{2}g}{t^{2}}=\frac{2\left(S-tu\right)}{t^{2}}$$

Dividing by \frac{1}{2}t^{2} undoes the multiplication by \frac{1}{2}t^{2}.

$$g=\frac{2\left(S-tu\right)}{t^{2}}$$

Solve for S

$$S=\frac{t\left(gt+2u\right)}{2}$$