The field of elimination theory was motivated by the need of methods for solving systems of polynomial equations.
The case of linear equations was completely solved by Gaussian elimination, where the older method of Cramer's rule does not proceed by elimination, and works only when the number of equations equals the number of variables.
Before the 20th century, different types of eliminants were introduced, including resultants, and various kinds of discriminants.
Later, elimination theory was considered old-fashioned and removed from subsequent editions of Moderne Algebra.
The main methods for this renewal of elimination theory are Gröbner bases and cylindrical algebraic decomposition, introduced around 1970.