Solve y=y_{0}+1/2(v+v_{o})t

$$y = y _ { 0 } + \frac { 1 } { 2 } ( v + v _ { o } ) t$$

Solve for t

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

Steps for Solving Linear Equation

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

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

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

$$y=y_{0}+\frac{1}{2}vt+\frac{1}{2}v_{o}t$$

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

$$y_{0}+\frac{1}{2}vt+\frac{1}{2}v_{o}t=y$$

Subtract y_{0} from both sides.

$$\frac{1}{2}vt+\frac{1}{2}v_{o}t=y-y_{0}$$

Combine all terms containing t.

$$\left(\frac{1}{2}v+\frac{1}{2}v_{o}\right)t=y-y_{0}$$

The equation is in standard form.

$$\frac{v+v_{o}}{2}t=y-y_{0}$$

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

$$\frac{2\times \frac{v+v_{o}}{2}t}{v+v_{o}}=\frac{2\left(y-y_{0}\right)}{v+v_{o}}$$

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

$$t=\frac{2\left(y-y_{0}\right)}{v+v_{o}}$$

Solve for v

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

Steps for Solving Linear Equation

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

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

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

$$y=y_{0}+\frac{1}{2}vt+\frac{1}{2}v_{o}t$$

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

$$y_{0}+\frac{1}{2}vt+\frac{1}{2}v_{o}t=y$$

Subtract y_{0} from both sides.

$$\frac{1}{2}vt+\frac{1}{2}v_{o}t=y-y_{0}$$

Subtract \frac{1}{2}v_{o}t from both sides.

$$\frac{1}{2}vt=y-y_{0}-\frac{1}{2}v_{o}t$$

The equation is in standard form.

$$\frac{t}{2}v=-\frac{tv_{o}}{2}+y-y_{0}$$

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

$$\frac{2\times \frac{t}{2}v}{t}=\frac{2\left(-\frac{tv_{o}}{2}+y-y_{0}\right)}{t}$$

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

$$v=\frac{2\left(-\frac{tv_{o}}{2}+y-y_{0}\right)}{t}$$

Divide y-\frac{v_{o}t}{2}-y_{0} by \frac{1}{2}t.

$$v=\frac{2y-2y_{0}-tv_{o}}{t}$$