SOLUTION: graph using the slope intercept a) y=3x+1 b) 3x-4y=12

I'll do the second one first, since it requires more work,
then I'll do the first one:

Solve it for y:

3x - 4y = 12
    -4y = -3x + 12

Divide through by 4

    y = x + 

          y = x - 3

compare that to 

          y = mx + b

the slope m is  and the y-intercept is (0,b) or (0,-3)

We begin at the y-intercept (0,-3):


         
We take the numerator of the slope which is 3, and since the
slope is positive we draw a line from the y-intercept UPWARD
3 units, like the green arrow below. [If the slope had been
negative, which it wasn't, we would have draw the arrow downward]



Now we take the denominator of the slope which is 4, and draw a 
another line from that green arrow head to the RIGHT 4 units, as 
shown below. [Even if the slope had been negative, we still would have 
drawn the arrow to the RIGHT the number of units indicated by the
denominator of the slope.]




Now get a ruler or straight edge and draw a straight line
through the y-intercept (0,-3) and through the second arrow head:



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

Now I'll do the first one:

          y = 3x + 1

It is already solved for y and is in slope-y-intercept form,
so we don't have to do that like we did in the other one:

compare that to 

          y = mx + b

the slope m is 3 and the y-intercept is (0,b) or (0,1)

As before we begin at the y-intercept (0,1):


        
Now the slope 3 in this one doesn't have a visible denominator,
so we write it as  so it will have both a numerator and
a denominator:  

We take the numerator of the slope which is 3, and since the
slope is positive we draw a line from the y-intercept UPWARD
3 units, like the green arrow below. [If the slope had been
negative, which it wasn't, we would have draw the arrow downward]



Now we take the denominator of the slope which is 1, and draw a 
another line from that green arrow head to the RIGHT 1 unit, as 
shown below. [Even if the slope had been negative, we still would have 
drawn the arrow to the RIGHT the number of units indicated by the
denominator of the slope.]




Now get a ruler or straight edge and draw a straight line
through the y-intercept (0,1) and through the second arrow head:




Edwin