Solve r+x=ru+varepsilon

$$r + x = r u + \varepsilon$$

Solve for r (complex solution)

$$\left\{\begin{matrix}r=-\frac{x-\epsilon }{1-u}\text{, }&u\neq 1\\r\in \mathrm{C}\text{, }&x=\epsilon \text{ and }u=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ru from both sides.

$$r+x-ru=\epsilon$$

Subtract x from both sides.

$$r-ru=\epsilon -x$$

Combine all terms containing r.

$$\left(1-u\right)r=\epsilon -x$$

Divide both sides by 1-u.

$$\frac{\left(1-u\right)r}{1-u}=\frac{\epsilon -x}{1-u}$$

Dividing by 1-u undoes the multiplication by 1-u.

$$r=\frac{\epsilon -x}{1-u}$$

Solve for u (complex solution)

$$\left\{\begin{matrix}u=-\frac{-x+\epsilon -r}{r}\text{, }&r\neq 0\\u\in \mathrm{C}\text{, }&r=0\text{ and }x=\epsilon \end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ru+\epsilon =r+x$$

Subtract \epsilon from both sides.

$$ru=r+x-\epsilon$$

The equation is in standard form.

$$ru=x+r-\epsilon$$

Divide both sides by r.

$$\frac{ru}{r}=\frac{x+r-\epsilon }{r}$$

Dividing by r undoes the multiplication by r.

$$u=\frac{x+r-\epsilon }{r}$$

Solve for r

$$\left\{\begin{matrix}r=-\frac{x-\epsilon }{1-u}\text{, }&u\neq 1\\r\in \mathrm{R}\text{, }&x=\epsilon \text{ and }u=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ru from both sides.

$$r+x-ru=\epsilon$$

Subtract x from both sides.

$$r-ru=\epsilon -x$$

Combine all terms containing r.

$$\left(1-u\right)r=\epsilon -x$$

Divide both sides by 1-u.

$$\frac{\left(1-u\right)r}{1-u}=\frac{\epsilon -x}{1-u}$$

Dividing by 1-u undoes the multiplication by 1-u.

$$r=\frac{\epsilon -x}{1-u}$$

Solve for u

$$\left\{\begin{matrix}u=-\frac{-x+\epsilon -r}{r}\text{, }&r\neq 0\\u\in \mathrm{R}\text{, }&r=0\text{ and }x=\epsilon \end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ru+\epsilon =r+x$$

Subtract \epsilon from both sides.

$$ru=r+x-\epsilon$$

The equation is in standard form.

$$ru=x+r-\epsilon$$

Divide both sides by r.

$$\frac{ru}{r}=\frac{x+r-\epsilon }{r}$$

Dividing by r undoes the multiplication by r.

$$u=\frac{x+r-\epsilon }{r}$$