You want to isolate y on one side, and the rest on the other. 

Start by adding 7y to both sides to get  6 + 4x/3 = 1 + 7y

Then subtract 1 from both sides to get 4x/3 + 5 = 7

Now, divide both sides by 7. (4x/3)/(7) + (5)/(7) = y

Simplifying, we get 4x/21 + 5/7 = y

Wait... doesn't y belong on the left? We usually see it that way, but the Symmetric Property of Equality says that if a = b then b = a. Therefore, we can rewrite the equation as y = 4x/21 + 5/7

The slope-intercept form is y=mx+b, so you have to solve given equation for y.
 6 - 7y + 4x/3 = 1
-6                    -6
    - 7y + 4x/3 = -5
            - 4x/3   - 4x/3
     - 7y = -4x/3 - 5
     -7y/(-7) = (-4x/3)/(-7) -5/(-7)
       y = (4/21)x + 5/7
4/21 is a slope, and (0 , 5/7) is coordinates of y-intercept