Solve y=v_{0}yt-1/2gt^2

$$y = v _ { 0 } y t - \frac { 1 } { 2 } g t ^ { 2 }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=\frac{2y\left(tv_{0}-1\right)}{t^{2}}\text{, }&t\neq 0\\g\in \mathrm{C}\text{, }&y=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v_{0}yt-\frac{1}{2}gt^{2}=y$$

Subtract v_{0}yt from both sides.

$$-\frac{1}{2}gt^{2}=y-v_{0}yt$$

The equation is in standard form.

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

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

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

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

$$g=\frac{y-tv_{0}y}{-\frac{t^{2}}{2}}$$

Divide y-yv_{0}t by -\frac{1}{2}t^{2}.

$$g=-\frac{2y\left(1-tv_{0}\right)}{t^{2}}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{2y\left(tv_{0}-1\right)}{t^{2}}\text{, }&t\neq 0\\g\in \mathrm{R}\text{, }&y=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v_{0}yt-\frac{1}{2}gt^{2}=y$$

Subtract v_{0}yt from both sides.

$$-\frac{1}{2}gt^{2}=y-v_{0}yt$$

The equation is in standard form.

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

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

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

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

$$g=\frac{y-tv_{0}y}{-\frac{t^{2}}{2}}$$

Divide y-ytv_{0} by -\frac{1}{2}t^{2}.

$$g=-\frac{2y\left(1-tv_{0}\right)}{t^{2}}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{\sqrt{y\left(yv_{0}^{2}-2g\right)}+v_{0}y}{g}\text{; }t=\frac{-\sqrt{y\left(yv_{0}^{2}-2g\right)}+v_{0}y}{g}\text{, }&g\neq 0\\t=\frac{1}{v_{0}}\text{, }&g=0\text{ and }y\neq 0\text{ and }v_{0}\neq 0\\t\in \mathrm{C}\text{, }&g=0\text{ and }y=0\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=\frac{\sqrt{y\left(yv_{0}^{2}-2g\right)}+v_{0}y}{g}\text{; }t=\frac{-\sqrt{y\left(yv_{0}^{2}-2g\right)}+v_{0}y}{g}\text{, }&\left(v_{0}=0\text{ or }y\geq 0\text{ or }y\leq \frac{2g}{v_{0}^{2}}\right)\text{ and }\left(v_{0}=0\text{ or }y\leq 0\text{ or }y\geq \frac{2g}{v_{0}^{2}}\right)\text{ and }g\neq 0\\t=\frac{1}{v_{0}}\text{, }&g=0\text{ and }y\neq 0\text{ and }v_{0}\neq 0\\t\in \mathrm{R}\text{, }&g=0\text{ and }y=0\end{matrix}\right.$$