Describe vectors v = [x , y] that are orthogonal to u = [a , b].

Question

The correct answer is: $\mathbf{u}\cdot\mathbf{v}= a⋅x+b⋅y = 0\rightarrow y = −(a/b)x\rightarrow \mathbf{v}=(t,-\frac{a}{b}t)$. But how did they find $-(a/b)$ and why did they substitute $y$ with $-(a/b)x$?

Answer 1

$$ax+by=0 \Longrightarrow ax+by-ax=-ax \Longrightarrow by=-ax \Longrightarrow\frac{by}{b}=-\frac{a}{b}x\Longrightarrow y=-\frac{a}{b}x$$

I am assuming that $b\neq 0$.