Question 268805
How do you solve:

Write an equation of a line which contains the points (3, -2) and (3,6).

<pre><font size = 4 color = "indigo"><b>

Let's plot those two points:

{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10),

line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1), locate(3,-2,"(3,-2)"),

line(3+.1,6,3-.1,6), line(3,6+.1,3,6-.1), line(3+.1,6+.1,3-.1,6-.1), line(3+.1,6-.1,3-.1,6+.1), locate(3,6,"(3,6)")

)}}}

Now I will draw a green line through them:

{{{drawing(400,400,-10,10,-10,10,

graph(400,400,-10,10,-10,10), green(line(3,-11,3,11)),

line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1), locate(3,-2,"(3,-2)"),

line(3+.1,6,3-.1,6), line(3,6+.1,3,6-.1), line(3+.1,6+.1,3-.1,6-.1), line(3+.1,6-.1,3-.1,6+.1), locate(3,6,"(3,6)")

)}}}


Hmm! That's a very special type of line!

Notice that it is VERICAL.  Vertical lines are the only kinds of lines 
that DO NOT have slopes or y-intercepts!  

However, vertical lines DO have equations.  Notice that the 
two points you were given both have the same x-coordinate 3.

Look at some other points on that vertical line. Five more
points on that line are (3,5), (3,4), (3,2), (3,-3), and (3,-7):

{{{drawing(400,400,-10,10,-10,10, green(line(3,-11,3,11)),

line(3+.1,-2,3-.1,-2), line(3,-2+.1,3,-2-.1), line(3+.1,-2+.1,3-.1,-2-.1), line(3+.1,-2-.1,3-.1,-2+.1), locate(3,-2,"(3,-2)"),

line(3+.1,6,3-.1,6), line(3,6+.1,3,6-.1), line(3+.1,6+.1,3-.1,6-.1), line(3+.1,6-.1,3-.1,6+.1), locate(3,6,"(3,6)"),

line(3+.1,5,3-.1,5), line(3,5+.1,3,5-.1), line(3+.1,5+.1,3-.1,5-.1), line(3+.1,5-.1,3-.1,5+.1), locate(3,5,"(3,5)"),

line(3+.1,2,3-.1,2), line(3,2+.1,3,2-.1), line(3+.1,2+.1,3-.1,2-.1), line(3+.1,2-.1,3-.1,2+.1), locate(3,2,"(3,2)"),

line(3+.1,-7,3-.1,-7), line(3,-7+.1,3,-7-.1), line(3+.1,-7+.1,3-.1,-7-.1), line(3+.1,-7-.1,3-.1,-7+.1),  locate(3,-7,"(3,-7)"),
line(3+.1,4,3-.1,4), line(3,4+.1,3,4-.1), line(3+.1,4+.1,3-.1,4-.1), line(3+.1,4-.1,3-.1,4+.1), 

line(3+.1,-3,3-.1,-3), line(3,-3+.1,3,-3-.1), line(3+.1,-3+.1,3-.1,-3-.1), line(3+.1,-3-.1,3-.1,-3+.1),  locate(3,4,"(3,4)"), locate(3,-3,"(3,-3)"), 

graph(400,400,-10,10,-10,10)  )}}}

In fact, EVERY point on that line has its x-coordinate
as 3.  

So to describe that vertical green line, we could just say

"The x-coordinate of any point on the line always equals 3"

or

"x always equals 3"

or even shorter

"x = 3"

That's the way to describe a vertical line, just

write "x =" and put whatever number after it

that the x-coordinates of all the points on it are,

in this case all the x-coordinates are 3.

So the equation of that vertical line is

x = 3

You cannot put it in slope-intercept form, 
for two reasons:

1.  It has no slope!

and

2. It has no y-intercept!

So you just have to leave the equation as simply

x = 3

Note:
The problem you have submitted is an unusual kind of problem
because every other kind of line except a vertical line has
a slope and a y-intercept.  All vertical lines have equations 
of the form x = k 

Edwin</pre>