Question 156607
<font size = 7 color = "red"><b>Edwin's explanation:</b></pre></font>

Explain why the line x = 4 is a
vertical line.

<pre><font size = 4 color = "indigo"><b> 
Let's draw a vertical line through 4 on the x axis, like this:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
rectangle(4,-11,11,11) 
)}}}

Now let's plot a couple of points on that line, like this:


{{{drawing(400,400,-10,10,-10,10, locate(4-.2,5.4,"@"),
locate(4-.2,2.4,"@"), graph(400,400,-10,10,-10,10),
rectangle(4,-11,11,11) 
)}}}

They are the points (4,2) and (4,5). Let's label them:

{{{drawing(400,400,-10,10,-10,10, locate(4-.2,5.4,"@"),
locate(4-.2,2.4,"@"), graph(400,400,-10,10,-10,10),
rectangle(4,-11,11,11), locate(4.2,2.2,"(4,2)"), locate(4.2,5.2,"(4,5)") 
)}}}

Now let's get acouple more, down below the x-axis:

{{{drawing(400,400,-10,10,-10,10, locate(4-.2,5.4,"@"),
locate(4-.2,2.4,"@"), graph(400,400,-10,10,-10,10),
rectangle(4,-11,11,11), locate(4.2,2.2,"(4,2)"), locate(4.2,5.2,"(4,5)"),
locate(4-.2,-2.7,"@"),
locate(4-.2,-6.7,"@") 
)}}}

Those two are (4,-3) and (4,-7). Let's label them:

{{{drawing(400,400,-10,10,-10,10, locate(4-.2,5.4,"@"),
locate(4-.2,2.4,"@"), graph(400,400,-10,10,-10,10),
rectangle(4,-11,11,11), locate(4.2,2.2,"(4,2)"), locate(4.2,5.2,"(4,5)"),
locate(4-.2,-2.7,"@"),locate(4,-3,"(4,-3)"),locate(4,-7,"(4,-7)"), 
locate(4-.2,-6.7,"@") 
)}}}

Now look at all those points:

(4,2), (4,5), (4,-3), (4,-7)

What do you see that they have in common?  You will
notice that the x-coordinate of every one of them is 4.

Theoretically we could get thousands more points on that 
same line. We could plot (4,9), (4,-8), (4,0), (4,-1), 
etc., etc. If we drew the graph big enough we could even 
find the points (4,100), (4,-10000), (4,746), etc.,
etc.

So the best way to say 

"The line which has all the points which have 4 as their 
x-coordinate", is to say 

"the x-coordinate always equals 4 on this line".

or "x always equals 4", 

or just plain

x = 4.

------------------

Another way to look at it is to think of it this way:

      x = 4

is the same as
 
 x + 0y = 4

And if we make a table of points, just
picking arbitrary values for y, say,
2, 5, -3, and -7, (I picked the same
arbitrary ones I picked above)

x | y
------
  | 2
  | 5
  |-3
  |-7

and substitute those values of y into

   x + 0y = 4       x + 0y = 4        x + 0y = 4     x + 0y = 4
 x + 0(2) = 4     x + 0(5) = 4     x + 0(-3) = 4  x + 0(-7) = 4
    x + 0 = 4        x + 0 = 4         x + 0 = 4      x + 0 = 4
        x = 4            x = 4             x = 4          x = 4  

So the points are

x | y
------
4 | 2
4 | 5
4 |-3
4 |-7 

So if we plot those four points:

{{{drawing(400,400,-10,10,-10,10, locate(4-.2,5.4,"@"),
locate(4-.2,2.4,"@"), graph(400,400,-10,10,-10,10),
locate(4.2,2.2,"(4,2)"), locate(4.2,5.2,"(4,5)"),
locate(4-.2,-2.7,"@"),locate(4,-3,"(4,-3)"),locate(4,-7,"(4,-7)"),
locate(4-.2,-6.7,"@") 
)}}}

and then draw a straight line through them, we have:

{{{drawing(400,400,-10,10,-10,10, locate(4-.2,5.4,"@"),
locate(4-.2,2.4,"@"), graph(400,400,-10,10,-10,10),
rectangle(4,-11,11,11), locate(4.2,2.2,"(4,2)"), locate(4.2,5.2,"(4,5)"),
locate(4-.2,-2.7,"@"),locate(4,-3,"(4,-3)"),locate(4,-7,"(4,-7)"), 
locate(4-.2,-6.7,"@") 
)}}}

Do you think you understand it now why we say that vertical
line has the equation " x = 4 "?  (For that is a true statement
about every point on that line.  It's x-value or x-coordinate
is always equal to 4.

Edwin</pre>