We will change the slope first and then we will slide it down

In the equation:

y = x + 4, is of the form y = mx+b

The 4 tells us where the line crosses the y-axis.

the  tells us how steep the line is.  The numerator
of  is 1 and the denominator is 3.  That tells
us that the line goes up 1 unit for every 3 units it goes
to the right.   

So it looks like this:

    

You will notice in the right graph that the green lines show that it 
rises 1 unit for every 3 units it moves to the right. 

Now we want to change the slope by a factor of 3.  That is, we
want to make the line 3 times as steep.  So that instead of rising
only 1 unit for every 3 units it runs, we want it to rise 3 times
as many units for every 3 units it moves to the right. To do that
we will multiply the slope  by  making it 
or .  Yes I know that reduces to 1 but I am leaving the 
denominator as 3, to show that it is now 3 times as steep as the 
original line.  I haven't changed the 4 yet.  I will do that next
when I slide it down 5 units. Here is the graph of 

y = x + 4

     

And again the green lines on the right show that this new line rises
3 times as much per 3 units of movement to the right as the original
line did.  And now I will reduce the  to just 1 and the
equation of this new line is

y = 1x + 4 

or just 

y = x + 4

Now since the 4 tells us where the line crosses the y-axis, and we want
the line to slide down the y-axis by 5 units, we simply subtract 5
from the 4 getting -1.  So this is the final graph of

y = x - 1

    

In other word, take the equation

y = x + 4, multiply the slope  by 3 and subtract 5
from the 4:

y = 3·x + 4 - 5

y = x - 1

So you might ask why I didn't just say "multiply  by 3 
and subtract 5 from 4 in the beginning.
If so my answer is: because it is extremely important that you
understand what you are doing GRAPHICALLY!  It's not enough
just to say "multiply and subtract".  What makes math hard is
trying to memorize what to do and ignoring why you do it.  Once
you understand why you do what you do in math, math becomes
easy.  Memorizing a thousand rules of "what to do" without
understanding why you do them is almost impossible.

Edwin