The problem of being parallel to $\mathbf y$ axis and undefined slope

Question

We all know the general equation for line $d$ is: $d:y=ax+b$ I want you to suppose $d$ is parallel to $y$ axis. Now , if we rearrange our equation , $x $will be equal to: $$x=\frac{-b}{a}$$ It is clear that $a$ is undefined as line$d$ is parallel to $y$axis.But how $x$is equal to $-b$ over undefined?

Answer 1

By explicit equation we can’t describe lines parallel to $y$ axis. Indeed in the explicit form

$$y=mx+n$$

$m$ represents the angular coefficient which $\to \infty$ for a vertical line thus this case can’t be described in such way.

For the general case we can refer to the implicit form

$$ax+by+c=0$$

which always holds.

Answer 2

"the general equation for line $d$ is: $d:y=ax+b$" This is not accurate. That's the general equation for a linear function, meaning that for each value of $x$ we get a unique value of $y$. This means that it specifically does not describe vertical lines, since for vertical lines there is only a single value of $x$ which works, and for that $x$-value there are infinitely many values for $y$.

A general equation for a line in the plane is $$ ax + by = c $$ where $a$ and $b$ are not both equal to $0$.

Answer 3

What is the meaning of slope of a line? This means take two points on the line where $x$ co-ordinates differ by 1, and see how much the $y$ co-ordinates differ for those two points.

For example, if $(3,7)$ and $(4,23)$ are two points on the line the difference $23-7=16$ is the slope of the line connecting them.

In the case of a line parallel to the $y$-axis all the points have the same $x$ co-ordinate. So slope is undefined. The equation is $x=c$, the constant value of $x$-coordinate.