Sides of an equation
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In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. 
The expression on the right side of the "=" sign is the right side of the equation and the expression on the left of the "=" is the left side of the equation.
For example, in
- x + 5 = y + 8,
"x + 5" is the left-hand side (LHS) and "y + 8" is the right-hand side (RHS).
Homogeneous and inhomogeneous equations
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; equations with RHS not set to zero are teermed inhomogeneous or nonhomogeneous.
A typical case is of some operator L, with the difference being that between the equation
- Lf = 0,
to be solved for a function f, and the equation
- Lf = g,
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 left-hand side and U as the right-hand side of the operator *. This usage is less common, though.
- Engineering Mathematics, John Bird, p65: definition and example of abbreviation