Solve r(-Deltay+t)(t+dy)=+r

$$r ( - \Delta y + t ) ( t + d y ) = + r$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{-ty\Delta +t^{2}-1}{y\left(t-y\Delta \right)}\text{, }&y\neq 0\text{ and }\Delta \neq \frac{t}{y}\text{ and }t\neq y\Delta \\d\in \mathrm{R}\text{, }&\left(y=0\text{ and }|t|=1\right)\text{ or }r=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply r by \left(-\Delta \right)y+t.

$$\left(r\left(-\Delta \right)y+rt\right)\left(t+dy\right)=r$$

Use the distributive property to multiply r\left(-\Delta \right)y+rt by t+dy.

$$r\left(-\Delta \right)yt+r\left(-\Delta \right)dy^{2}+rt^{2}+rtdy=r$$

Subtract r\left(-\Delta \right)yt from both sides.

$$r\left(-\Delta \right)dy^{2}+rt^{2}+rtdy=r-r\left(-\Delta \right)yt$$

Subtract rt^{2} from both sides.

$$r\left(-\Delta \right)dy^{2}+rtdy=r-r\left(-\Delta \right)yt-rt^{2}$$

Multiply -1 and -1 to get 1.

$$r\left(-1\right)\Delta dy^{2}+rtdy=r+r\Delta yt-rt^{2}$$

Combine all terms containing d.

$$\left(r\left(-1\right)\Delta y^{2}+rty\right)d=r+r\Delta yt-rt^{2}$$

The equation is in standard form.

$$\left(rty-r\Delta y^{2}\right)d=rty\Delta -rt^{2}+r$$

Divide both sides by -r\Delta y^{2}+rty.

$$\frac{\left(rty-r\Delta y^{2}\right)d}{rty-r\Delta y^{2}}=\frac{r\left(ty\Delta -t^{2}+1\right)}{rty-r\Delta y^{2}}$$

Dividing by -r\Delta y^{2}+rty undoes the multiplication by -r\Delta y^{2}+rty.

$$d=\frac{r\left(ty\Delta -t^{2}+1\right)}{rty-r\Delta y^{2}}$$

Divide r\left(1+\Delta yt-t^{2}\right) by -r\Delta y^{2}+rty.

$$d=\frac{ty\Delta -t^{2}+1}{y\left(t-y\Delta \right)}$$