I don't agree that all lines have a y-intercept. After all, it is possible to have a line that is vertical and parallel to the y-axis. In such a case, the line would not cross the axis and would have no intercept.
Granted, this line cannot be described in a y = mx + b equation. It has to be defined simply as (for example) x = 2. But still, such a line can exist and it would be graphed as a vertical line that intersects the x axis at (2,0).
Not all the lines have x intercepts.
For instance, a constant function f(x)=c is represented by a line that is parallel to x axis, never intercepting it.
Instead, all lines have y intercepts.
We'll have an example
y = mx+n
m is the slope of the line and n is y intercept
We'll calculate x intercept:
y = 0 <=> mx+n = 0 <=> x = -m/n
The intercepting point is (-m/n , 0)
We'll calculate y intercept:
x = 0 <=> y=m*0+n <=> y=n
The intercepting point is (0 , n).