Here's a more detailed explanation of what the other tutor was saying.

Let's plot those two points:



Now I will draw a green line through them:



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

Notice that it is NOT slanted as you said above. 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 6.

Look at some other points on that vertical line. Three more
points on that line are (6,5) and (6,2), and (6,-7):


In fact, EVERY point on that line has its x-coordinate
as 6.  

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

"The x-coordinate of any point on the line always equals 6"

or

"x always equals 6"

or even shorter

"x = 6"

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 6.

So the equation of that vertical line is

x = 6

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 = 6

You mentioned "the perpendicular equation".  There
is no one line to call "THE perpendicular line" or
"THE perpendicular equation".  That are many many
perpendicular lines to any given line, and every one
of them has a different equation.  

Now any line perpendicular to the green vertical line
is horizontal.  For instance the red vertical line below:



It goes through the point (6,4) and in fact every point on
that red horizontal line has 4 as its y-coordinate.  For
instance look at three more points on that horizontal line






So to describe that horizontal red line, we could just say

"The y-coordinate of any point on the red line always equals 4"

or

"y always equals 4"

or even shorter

"y = 4"

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

write "y =" and put whatever number after it

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

in this case 4.

However, unlike vertical lines, horizontal lines DO have a slope,
and a y intercept.  You will notice that the red horizontal line
above crosses the y-axis at 4 so it has the y-intercept 4.  Its
slope is the number 0.  So its slope-intecept form is

y = 0x + 4 which is a form of y = mx + b.  Since the slope is
0 the term "0x" is usually not written but it is understood. 

Every horizontal line is perpendicular to the green line above.

Every one of them has an equation of the form 

y = 0x + k 

where k represents the number at which the horizontal line 
crosses the y-axis, i.e., its y-intercept.

I have a hunch that what your teacher means by "the perpendicular
equation" is simply 

y = k

where k can represent any number.  (Or maybe your teacher
would prefer y = 0x + b, where 0 is the slope and b is
the y-coordinate of the y-intercepr).  

Note:
The problem you have submitted is an unusual kind of problem
because every other kind of line exept a vertical line has
a slope and a y-intercept.  Only vertical lines have neither
slopes nor y-intercepts.  As you saw above even horizontal
lines have a slope of 0 and also a y-intercept at wherever they
cross the y-axis.  Only vertical lines have neither slopes nor
y-intercepts, but their equation is always of the for x = k, where
k represents some number.  And also you noticed that all
horizontal lines have as their equation y = k or b, where k or b
represents the y-intercept as well as the y-cioordinate of every
point on the horizontal line.

Edwin