Sorry, my fingers are slower than my head.

Perpendicular lines slopes always share a negative inverse relationship.
     Mathematically this means that the product of the two slopes always equals negative one.

  m1(m2) = -1

     A linear equation can always be written as mx + b = y

     You were on the right track to solving this problem.

     Solve for y for the original equation:  y= -2/3 x -2/3

     Therefore the slope is -2/3

     If the second line is perpendicular then -2/3 m2 = -1,  so m2 = 3/2 or 1.5

     The second equation must then be y = 1.5x + b
   
     Plugging in the point specified by your question:  7 = 1.5(4) + b

     Solving for b and keeping the slope give you your perpendicular line equation

    y = 1.5x + 1

You got the gradient right, because to find a perpendicular gradient, you divide -1 by the original gradient.

If you know a line has gradient g and passes through (x,y), then its y-intercept is y - (g*x).

For your one, it passes through (4,7) and has a gradient of 3/2, so the y-intercept is 7 - (3/2*4) = 1.

I've completed a few of these with no problem but I am stuck on:
2x + 3y + 2 = 0,  find the equation of the perpendicular line passing through (4, 7).

2x + 3y + 2 = 0
3y = -2x - 2
y = (-2x -2)/3

perpendicular line would be :
y = 3/2 x + 4 1/3?