Solve t^2y-ct=-(x-ct)

$$t ^ { 2 } y - c t = - ( x - c t )$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=\frac{x+yt^{2}}{2t}\text{, }&t\neq 0\\c\in \mathrm{C}\text{, }&x=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To find the opposite of x-ct, find the opposite of each term.

$$t^{2}y-ct=-x+ct$$

Subtract ct from both sides.

$$t^{2}y-ct-ct=-x$$

Combine -ct and -ct to get -2ct.

$$t^{2}y-2ct=-x$$

Subtract t^{2}y from both sides.

$$-2ct=-x-t^{2}y$$

Reorder the terms.

$$-2ct=-x-yt^{2}$$

The equation is in standard form.

$$\left(-2t\right)c=-x-yt^{2}$$

Divide both sides by -2t.

$$\frac{\left(-2t\right)c}{-2t}=\frac{-x-yt^{2}}{-2t}$$

Dividing by -2t undoes the multiplication by -2t.

$$c=\frac{-x-yt^{2}}{-2t}$$

Divide -x-yt^{2} by -2t.

$$c=\frac{ty}{2}+\frac{x}{2t}$$

Solve for c

$$\left\{\begin{matrix}c=\frac{x+yt^{2}}{2t}\text{, }&t\neq 0\\c\in \mathrm{R}\text{, }&x=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To find the opposite of x-ct, find the opposite of each term.

$$t^{2}y-ct=-x+ct$$

Subtract ct from both sides.

$$t^{2}y-ct-ct=-x$$

Combine -ct and -ct to get -2ct.

$$t^{2}y-2ct=-x$$

Subtract t^{2}y from both sides.

$$-2ct=-x-t^{2}y$$

Reorder the terms.

$$-2ct=-x-yt^{2}$$

The equation is in standard form.

$$\left(-2t\right)c=-x-yt^{2}$$

Divide both sides by -2t.

$$\frac{\left(-2t\right)c}{-2t}=\frac{-x-yt^{2}}{-2t}$$

Dividing by -2t undoes the multiplication by -2t.

$$c=\frac{-x-yt^{2}}{-2t}$$

Divide -x-yt^{2} by -2t.

$$c=\frac{ty}{2}+\frac{x}{2t}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{\sqrt{c^{2}-xy}+c}{y}\text{; }t=\frac{-\sqrt{c^{2}-xy}+c}{y}\text{, }&y\neq 0\\t=\frac{x}{2c}\text{, }&y=0\text{ and }c\neq 0\\t\in \mathrm{C}\text{, }&y=0\text{ and }c=0\text{ and }x=0\end{matrix}\right.$$

Solve for t

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