Equations of a space

Why does a cartesian equation for a space not exist, or does it, and I'm just not aware of it? I suppose I'm overlooking some detail, but what? Is 4-dimensional coordinate system needed to define a 3- dimensional space in a general equation?

An equation by nature sets two things as being equal. That lowers the degrees of freedom in the variable set by one. For example, if you have a single dimension where x is any real value, you have one degree of freedom (x can be anything). When you set an equation in it, say 2x + 3 = 5, that reduces the degree of freedom by one, to zero; that is, x = 1. In a plane you have two degrees of freedom, and can reduce that by one by setting an equation there; e.g. y = x^2; that is true for x equals any real value, say a, as long as y equals a^2. In space you have three degrees of freedom but nearly any equation reduces that freedom from three to two, giving you a plane.

To get a solution with three degrees of freedom, you could start with four and limit that by one to three.. say, (x,y,z,t) in a 4D space and use the equation t = 0, which is true for any real values of x, y, and z, as long as t is zero.

alternatively.. you could use a non-conditional equation that does not reduce the degrees of freedom. Say, the values x,y,z for which

x + y + z = x + y + z.