Aus Online Mathematik Brückenkurs 2

If we multiply both sides by \displaystyle z+i, we avoid having \displaystyle z in the denominator,

\displaystyle iz+1=(3+i)(z+i)\,\textrm{.}

At the same time, this means that we are now working with a new equation which is not necessarily entirely equivalent with the original equation. Should the new equation show itself to have \displaystyle z=-i as a solution, then we must ignore that solution, because our initial equation cannot possibly have \displaystyle z=-i as a solution (the denominator of the left-hand side becomes zero).

We expand the right-hand side in the new equation,

\displaystyle iz+1 = 3z+3i+iz-1\,,

and move all the terms in \displaystyle z over to the left-hand side and the constants to the right-hand side,

\displaystyle \begin{align} iz-3z-iz &= 3i-1-1\,,\\[5pt] -3z &= -2+3i\,\textrm{.} \end{align}

Then, we obtain

\displaystyle z = \frac{-2+3i}{-3} = \frac{2}{3}-i\,\textrm{.}

It is a little troublesome to divide two complex numbers, so we will therefore not check whether \displaystyle z=\tfrac{2}{3}-i is a solution to the original equation, but satisfy ourselves with substituting into the equation \displaystyle iz+1=(3+i)(z+i),

\displaystyle \begin{align} \text{LHS} &= iz+1 = i(\tfrac{2}{3}-i)+1 = \tfrac{2}{3}\cdot i+1+1 = 2+\tfrac{2}{3}i,\\[5pt] \text{RHS} &= (3+i)(z+i) = (3+i)(\tfrac{2}{3}-i+i) = (3+i)\tfrac{2}{3} = 2+\tfrac{2}{3}i\,\textrm{.} \end{align}