Solve s=a/2*t+v_{0}*t+s_{0}

$$s = \frac { a } { 2 } \cdot t + v _ { 0 } \cdot t + s _ { 0 }$$

Solve for a

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

Steps for Solving Linear Equation

Express \frac{a}{2}t as a single fraction.

$$s=\frac{at}{2}+v_{0}t+s_{0}$$

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

$$\frac{at}{2}+v_{0}t+s_{0}=s$$

Subtract v_{0}t from both sides.

$$\frac{at}{2}+s_{0}=s-v_{0}t$$

Subtract s_{0} from both sides.

$$\frac{at}{2}=s-v_{0}t-s_{0}$$

Multiply both sides of the equation by 2.

$$at=2s-2v_{0}t-2s_{0}$$

The equation is in standard form.

$$ta=2s-2s_{0}-2tv_{0}$$

Divide both sides by t.

$$\frac{ta}{t}=\frac{2s-2s_{0}-2tv_{0}}{t}$$

Dividing by t undoes the multiplication by t.

$$a=\frac{2s-2s_{0}-2tv_{0}}{t}$$

Divide 2s-2v_{0}t-2s_{0} by t.

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

Solve for s

$$s=\frac{at}{2}+tv_{0}+s_{0}$$