Let’s start from a very simple equation:
#y = x#
From the graph of this equation, you can tell that it has a slope of #1# because it rises #1# for every #1# we run.
But we need a slope of #2#. This means that we need a line that rises #2# for every #1# you run. We have to double the steepness of our line.
To double the #y# for every #x# that we run, we change our equation to:
#y = 2x#
You can graph this equation to verify that it rises #2# for every #1# we run.
But this line does not pass through point #(3,1)#. We know that because, when #x# is #3#, #y# is #6# (not #1#).
By subtracting #5#, we will make every #y# coordinate of our line go down #5# units. And we do want that. So we try the equation:
# y = 2x - 5#
Let’s try this equation with #x=2#, #x=3#, and #x=4#.
Say #x=2#. Then #y = 2(2) - 5 = 4 - 5 = -1#
Say #x=3#. Then #y = 2(3) - 5 = 6 - 5 = 1#
Say #x=4#. Then #y = 2(4) - 5 = 8 - 5 = 3#
From this, we notice two things. The first one is that our line rises #2# for every #1# we run. So it still has a slope of #2#. The second one is that when #x=3#, #y=1#. This tells us that our line passes through #(3,1)#.
So, our answer is:
#y = 2x - 5#
