$xy=0$
$ax +by +cxy +d=0$
$ax +by +cz +dxy +eyz +gxyz=0$
I made myself the examples, sometimes I face these equations and I do not know how to resolve them, all equations whose unknowns have exponent equals one but they can be multiplied together as I have put in the example $xy$, $yxzt$.... I want to know the name so I can find info and understand them because I google equations and many different come, mostly linear but I do not see these ones. I did not see the tag "equations" so I tagged differential-equations but I do not think It is that.
In terms of nomenclature, equations of the form $$ c_1 x_1 + c_2 x_2 + \ldots $$
where $c$ is a constant are known as linear equations, equations: $$ c_1 x_1 x_2 + c_2 x_2 x_3 + \ldots $$
are bilinear equations. Though not as common, equations of the form: $$ c_1 x_1 x_2 x_3 + c_2 x_2 x_3 x_4 + \ldots $$ are referred to as trilinear equations. For higher orders @Raskolnikov suggested "systems of multilinear equations". Also, see the comment by @Lieven below who calls them "multilinear equations in homogeneous space".
Solving systems of linear equations is one of the goals of Linear Algebra. Less is known for bilinear equations, but such systems can often be "solved" in the sense of finding a near optimal solution (ie. as a constraint problem).