The Chap. 1, Sec. 2.3 in Complex Analysis by Lars V. Ahlfors writes
A straight line in the complex plane can be given by a parametric equation $z= a + b t$, where $a$ and $b$ are complex numbers and $b \neq 0$; the parameter $t$ runs through all real values. Two equations $z = a + b t$ and $z = a' + b' t$ represent the same line if and only if $a' - a$ and $b'$ are real multiples of $b$.
I find it difficult to understand it. How to conclude that these two equations represent the same line if and only if $a-a'$ and $b'$ are real multiples of $b$? My idea is that there exists a "bijection" between $z = a + b t$ and $z = a' + b' t$; that is, we can find a $t_0\in\mathbb{R}$, which satisfies that $$a+b(t-t_0)=a'+b't.$$ And then? Or are there any other ways to explain this statement?
Thinking of complex numbers as plane vectors the real line $$ a+bt\qquad a, b\in\mathbb{C}, t\in\mathbb{R} $$ is the line through $a$ and parallel to $b$. Thus if the lines $$ a+bt,\qquad a^\prime+b^\prime t $$ coincide the two "direction" vectors $b$ and $b^\prime$ must be parallel, i.e. one a real multiple of the other.
So, after reparametrization, the second line can be written $$ a^\prime+bu,\qquad u\in\mathbb{R}. $$ Again. the two line coincide, thus we must have a coincidence of sets of numbers $$ a+bt=a^\prime+bu $$ as $t$ and $u$ vary in $\mathbb{R}$. But this means that for some $t$ and $u$ (and we can be assume $t\neq u$ or else the result becomes trivial) we have an equality of numbers $$ a-a^\prime = b(u-t) $$ which is exactly what we want.