Linear Functions

f[x_, y_] := m x  +  n y  + c

Give some arbitrary values to the parameters m,n,and c:

m = 1; n = -1; c = 2;

Draw some cross-sections by setting x=constant, and then y=constant:

Plot[{f[0, y], f[1, y], f[2, y]}, {y, -5, 
          5}];
Plot[{f[x, 0], f[x, 1], f[x, 2]}, {x, -5, 5}];

Then combine these (and other) cross-sections to make the surface f[x,y]

Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}];

Equation of the plane passing through three points

Clear[m, n, c]

f[x_, y_] := m x  +  n y  + c

Suppose we are given the three points (0,1,0); (1,0,5); and (1,1,3); what is the equation of the linear plane passing through these points? Solve as a set of simultaneous equations.

Solve[{f[0, 1] == 0, f[1, 0] == 5, f[1, 1] == 3}, {m, 
        n, c}]

To find the equation representing contours of f[x,y], we let f[x,y]= constant = k (say), and then solve for y

Solve[f[x, y] == k, y]