How to express equation of line in set builder form?

I found set builder form of writing equation of straight line like $\left\{(8t+8,2t+5)∣t∈R\right\}$ on this answer. I found this way of the equation of line unique. I'm seeing this first time and interested in knowing how to convert any linear equation of the form $ax + by + c =0 $ in the set builder notations like $\left\{(8t+8,2t+5)∣t∈R\right\}$. What's the basic idea of representing equations in this form?

I would be grateful if anyone could tell me how to convert the linear equation of a straight line in set builder notations. I searched well on the internet but could not find any good article on this topic. Neither do our schools teach this way of writing equations.

What you are asking for is a parametric equation of a line, expressed in set-builder notation. To start, the line $3x + 2y - 6 = 0$ can be written in set-builder notation as $\{(x,y)\,|\,\forall x\forall y \in \mathbb{R} \land y = -\frac{3}{2}x + 3\}$.

In the example, you can modify $(x,y)$ to be functions of $t$ such that $(x,y) = (2t, -3t + 3)$. There are a lot of ways to express $(x,y)$ into $(f(t), g(t))$, especially when $y = f(x)$ represents a line.

In general, it is of the form $\left\{\big[f(t), g(t)\big]\,\Big|\,t\in\mathbb{R}\right\}$ where $(x,y) = (f(t), g(t))$ and expressing the parametric equation into the form $y = f(x)$ should yield a linear equation in terms of $x$.