Solve S=1/2(v_{0}+v)*t

$$S = \frac { 1 } { 2 } ( v _ { 0 } + v ) \cdot t$$

Solve for t

$$\left\{\begin{matrix}t=\frac{2S}{v+v_{0}}\text{, }&v_{0}\neq -v\\t\in \mathrm{R}\text{, }&S=0\text{ and }v_{0}=-v\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by v_{0}+v.

$$S=\left(\frac{1}{2}v_{0}+\frac{1}{2}v\right)t$$

Use the distributive property to multiply \frac{1}{2}v_{0}+\frac{1}{2}v by t.

$$S=\frac{1}{2}v_{0}t+\frac{1}{2}vt$$

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

$$\frac{1}{2}v_{0}t+\frac{1}{2}vt=S$$

Combine all terms containing t.

$$\left(\frac{1}{2}v_{0}+\frac{1}{2}v\right)t=S$$

The equation is in standard form.

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

Divide both sides by \frac{1}{2}v_{0}+\frac{1}{2}v.

$$\frac{2\times \frac{v+v_{0}}{2}t}{v+v_{0}}=\frac{2S}{v+v_{0}}$$

Dividing by \frac{1}{2}v_{0}+\frac{1}{2}v undoes the multiplication by \frac{1}{2}v_{0}+\frac{1}{2}v.

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

Solve for S

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