find a formula for a function g( x,y,z) whose level surfaces are planes parallel to z=3x-2y+7

To find a formula for the function g(x, y, z) with level surfaces parallel to the plane z = 3x - 2y + 7, we can start by considering the equation of a plane.

The equation of a plane can generally be written as ax + by + cz + d = 0, where (a, b, c) is the normal vector to the plane. In our case, the normal vector is (3, -2, 1) since the plane parallel to z = 3x - 2y + 7.

To obtain level surfaces parallel to this plane, we need to find a function g(x, y, z) such that g(x, y, z) = c, where c is a constant. This would give us a level surface at a specific height (c) in the z-direction parallel to the plane z = 3x - 2y + 7.

Let's rewrite the equation of the plane z = 3x - 2y + 7 in the form ax + by - z + d = 0 by multiplying by -1:
-3x + 2y - z + 7 = 0

Now, we can use this equation to obtain the formula for g(x, y, z) by rearranging the terms:
g(x, y, z) = 3x - 2y + z - 7

So, the formula for the function g(x, y, z), whose level surfaces are planes parallel to z = 3x - 2y + 7, is g(x, y, z) = 3x - 2y + z - 7.