= Solution
Write a <straight line> in Cartesian form $Ax+By+C=0$, where $A,B,C$ are real and $(A,B)\ne(0,0)$. With $z=x+iy$, set $c=(A+iB)/2$ and $r=C$. Using the <complex conjugate>,
$$
\overline c z+c\overline z=2\operatorname{Re}(\overline c z)=Ax+By.
$$
Thus the <real linear equation of a line in the complex plane> is
$$
\boxed{\overline c z+c\overline z+r=0,\qquad c\ne0,\quad r\in\mathbb R.}
$$
Conversely, writing $c=u+iv$ gives $2ux+2vy+r=0$, an ordinary <straight line> with normal <vector> $(u,v)$. The nonzero condition on $c$ matters: $c=0$ would give either the whole <complex plane> or the empty set.
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