In mathematics, LHS is informal shorthand for the lefthand side of an equation. Similarly, RHS is the righthand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric. This abbreviation is seldom if ever used in print; it is very informal.
More generally, these terms may apply to an inequation or inequality. In the inequality case, there is no symmetry. The righthand side is everything on the right side of a test operator in an expression. Conversely, the lefthand side is everything on the left side.
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The equation on the right side/right part of the "=" sign is the right side of the equation and the left of the "=" is the left side/left side of equation.
In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with the RHS set equal to zero. The corresponding inhomogeneous or nonhomogeneous equation then has the RHS with some given data, but of a general character.
The typical case is of some operator L, with the difference being that between the equation
to be solved for a function f, and the equation
with g a fixed function, to solve again for f. The point of the terminology appears for L a linear operator. Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution.
For example in mathematical physics, the homogeneous equation may correspond to a physical theory formulated in empty space, while the inhomogeneous equation asks for more 'realistic' solutions with some matter, or charged particles.
More abstractly, when using infix notation
the term T stands as the lefthand side and U as the righthand side of the operator *. This usage is less common, though.
