Definition of a scalar equation? What is a scalar equation? I have googled it and the only think that has come up is the equation of a plane
$$
ax+by+cz+d=0
$$
Is that the definition of a scalar equation? 
 A: As I said in the comment, the scalar equation should be something that is hardly context-dependent. You could choose to define the scalar equation as an equation that doesn`t admit vector solutions. 
But again, there are problems even with such a view, if you, for instance, choose your equation of the plane $ax+by+cz+d=0$ as some kind of scalar equation, then the solutions of the "equation of the plane" are triples $(x,y,z)$ of numbers and those triples can be seen as a vectors with the well-known laws of adding such triples and multiplying them with a scalar(huh?).
Even the scalars, if you take them to be real numbers, can be seen as a vectors because the field of real numbers can be seen as a vector space over the field of real numbers.
You could then try to define the scalar equation as an equation that has as the solution a scalar field, and it could be some partial differential equation which has scalar field as a solution, and then you´re maybe not left with an issues(unless you define addition of vector fields and multiplying them with a scalar so that they form a vector space) so I think you can choose this approach that scalar equations are solutions of partial differential equations which are not vectorial partial differential equations which have vector fields as the solutions.
A: I am guessing that a scalar equation is an equation which relates scalar quantities, as opposed to an equation which relates say functions, matrices, vectors, etc.. 
A: But
$$F= ma$$
with $a$ being acceleration ; it is vector quantity!
I think equations that show only loss or gain of quantities are scalar equations.
In scalars +,- do not stand for direction; these only show gain or loss.
An equation that is not concerned with direction of the quantities is a scalar equation.
