Solve t^r-1/t-1i+sqrt{rt^r-1}j+rtkt=1

$$\frac{ { t }^{ r } -1 }{ t-1 } i+ \sqrt{ r { t }^{ r } -1 } j+rtkt = 1$$

Solve for j

$$\left\{\begin{matrix}j=-\frac{\left(rt^{r}-1\right)^{-\frac{1}{2}}\left(krt^{3}-krt^{2}-t+it^{r}+\left(1-i\right)\right)}{t-1}\text{, }&t\neq 1\text{ and }\left(r=0\text{ or }\nexists n_{1}\in \mathrm{Z}\text{ : }t=e^{-\frac{2\pi n_{1}iRe(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}-\frac{2\pi n_{1}Im(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}+\frac{arg(\frac{1}{r})Im(r)+iarg(\frac{1}{r})Re(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}}\left(|r|\right)^{\frac{-Re(r)+iIm(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}}\right)\\j\in \mathrm{C}\text{, }&k=-\frac{1-i+it^{r}-t}{r\left(t-1\right)t^{2}}\text{ and }t\neq 0\text{ and }t\neq 1\text{ and }r\neq 0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }t=e^{-\frac{2iRe(r)\pi n_{1}}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}-\frac{2\pi n_{1}Im(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}+\frac{arg(\frac{1}{r})Im(r)+iarg(\frac{1}{r})Re(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}}\left(|r|\right)^{\frac{-Re(r)+iIm(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by t-1.

$$\left(t^{r}-1\right)i+\sqrt{rt^{r}-1}j\left(t-1\right)+rtkt\left(t-1\right)=t-1$$

Multiply t and t to get t^{2}.

$$\left(t^{r}-1\right)i+\sqrt{rt^{r}-1}j\left(t-1\right)+rt^{2}k\left(t-1\right)=t-1$$

Use the distributive property to multiply t^{r}-1 by i.

$$it^{r}-i+\sqrt{rt^{r}-1}j\left(t-1\right)+rt^{2}k\left(t-1\right)=t-1$$

Use the distributive property to multiply \sqrt{rt^{r}-1}j by t-1.

$$it^{r}-i+\sqrt{rt^{r}-1}jt-\sqrt{rt^{r}-1}j+rt^{2}k\left(t-1\right)=t-1$$

Use the distributive property to multiply rt^{2}k by t-1.

$$it^{r}-i+\sqrt{rt^{r}-1}jt-\sqrt{rt^{r}-1}j+rkt^{3}-rt^{2}k=t-1$$

Subtract it^{r} from both sides.

$$-i+\sqrt{rt^{r}-1}jt-\sqrt{rt^{r}-1}j+rkt^{3}-rt^{2}k=t-1-it^{r}$$

Add i to both sides.

$$\sqrt{rt^{r}-1}jt-\sqrt{rt^{r}-1}j+rkt^{3}-rt^{2}k=t-1-it^{r}+i$$

Subtract rkt^{3} from both sides.

$$\sqrt{rt^{r}-1}jt-\sqrt{rt^{r}-1}j-rt^{2}k=t-1-it^{r}+i-rkt^{3}$$

Add rt^{2}k to both sides.

$$\sqrt{rt^{r}-1}jt-\sqrt{rt^{r}-1}j=t-1-it^{r}+i-rkt^{3}+rt^{2}k$$

Reorder the terms.

$$jt\sqrt{rt^{r}-1}-j\sqrt{rt^{r}-1}=-krt^{3}+krt^{2}+t-it^{r}-1+i$$

Combine all terms containing j.

$$\left(t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}\right)j=-krt^{3}+krt^{2}+t-it^{r}-1+i$$

The equation is in standard form.

$$\left(t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}\right)j=-1+i-it^{r}+t+krt^{2}-krt^{3}$$

Divide both sides by t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}.

$$\frac{\left(t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}\right)j}{t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}}=\frac{-1+i-it^{r}+t+krt^{2}-krt^{3}}{t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}}$$

Dividing by t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1} undoes the multiplication by t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}.

$$j=\frac{-1+i-it^{r}+t+krt^{2}-krt^{3}}{t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}}$$

Divide -it^{r}-krt^{3}+krt^{2}+t+\left(-1+i\right) by t\sqrt{rt^{r}-1}-\sqrt{rt^{r}-1}.

$$j=\frac{\left(rt^{r}-1\right)^{-\frac{1}{2}}\left(-1+i-it^{r}+t+krt^{2}-krt^{3}\right)}{t-1}$$