Solve z=v*sin(x)*t-x/2t^2+y

$$z = v \cdot \sin ( x ) \cdot t - \frac { x } { 2 } t ^ { 2 } + y$$

Solve for v (complex solution)

$$\left\{\begin{matrix}v=-\frac{-xt^{2}+2y-2z}{2t\sin(x)}\text{, }&t\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\\v\in \mathrm{C}\text{, }&\left(z=-\frac{xt^{2}}{2}+y\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\right)\text{ or }\left(z=y\text{ and }t=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$z=v\sin(x)t-\frac{xt^{2}}{2}+y$$

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

$$v\sin(x)t-\frac{xt^{2}}{2}+y=z$$

Add \frac{xt^{2}}{2} to both sides.

$$v\sin(x)t+y=z+\frac{xt^{2}}{2}$$

Subtract y from both sides.

$$v\sin(x)t=z+\frac{xt^{2}}{2}-y$$

Multiply both sides of the equation by 2.

$$2v\sin(x)t=2z+xt^{2}-2y$$

The equation is in standard form.

$$2t\sin(x)v=xt^{2}-2y+2z$$

Divide both sides by 2\sin(x)t.

$$\frac{2t\sin(x)v}{2t\sin(x)}=\frac{xt^{2}-2y+2z}{2t\sin(x)}$$

Dividing by 2\sin(x)t undoes the multiplication by 2\sin(x)t.

$$v=\frac{xt^{2}-2y+2z}{2t\sin(x)}$$

Solve for v

$$\left\{\begin{matrix}v=-\frac{-xt^{2}+2y-2z}{2t\sin(x)}\text{, }&t\neq 0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\\v\in \mathrm{R}\text{, }&\left(y=z\text{ and }t=0\right)\text{ or }\left(y=\frac{xt^{2}}{2}+z\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$z=v\sin(x)t-\frac{xt^{2}}{2}+y$$

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

$$v\sin(x)t-\frac{xt^{2}}{2}+y=z$$

Add \frac{xt^{2}}{2} to both sides.

$$v\sin(x)t+y=z+\frac{xt^{2}}{2}$$

Subtract y from both sides.

$$v\sin(x)t=z+\frac{xt^{2}}{2}-y$$

Multiply both sides of the equation by 2.

$$2v\sin(x)t=2z+xt^{2}-2y$$

The equation is in standard form.

$$2t\sin(x)v=xt^{2}-2y+2z$$

Divide both sides by 2\sin(x)t.

$$\frac{2t\sin(x)v}{2t\sin(x)}=\frac{xt^{2}-2y+2z}{2t\sin(x)}$$

Dividing by 2\sin(x)t undoes the multiplication by 2\sin(x)t.

$$v=\frac{xt^{2}-2y+2z}{2t\sin(x)}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=-\frac{\sqrt{2\left(-v^{2}\cos(2x)+4xy-4xz+v^{2}\right)}-2v\sin(x)}{2x}\text{; }t=-\frac{-\sqrt{2\left(-v^{2}\cos(2x)+4xy-4xz+v^{2}\right)}-2v\sin(x)}{2x}\text{, }&x\neq 0\\t\in \mathrm{C}\text{, }&x=0\text{ and }z=y\end{matrix}\right.$$