Based on three points in collinear pairs

Suppose we know that a plane contains the three points: A = (3,2,2), B = (3,1,0), and C = (-1,2,5). How can we find a formula for the plane?

The key here is that the points are pairwise collinear. That is, there are two points with the same x value (A and B) and two points with the same y value (A and C). From the points A and B, we know that $\Delta x = 0$. If we connect these two points, we are looking at one of the sections with x = 3. The slope of this section is $\Delta z/\Delta y = -2/-1 = 2$. This is constant for all of the sections with x fixed at any value. Thus, if we assume the functional form z = mx + ny + b, we have just found the value of n in the formula.

Similarly, using points A and C, we get $m = \Delta z/\Delta x = 3/-4 = -3/4$. We now have two of the three constants in the formula. How can we find the third constant, b? Use one of the points we know, say B, and plug it into the know part of the formula:

$$0 = -(3/4)(3) + 2(1) + b \Rightarrow b = 1/4.$$

Thus, the formula for this plane is

$$z = -\frac{3}{4}x + 2y + \frac{1}{4}.$$