elimination of unknown
Consider the simultaneous polynomial equations
in two unknowns and , where e.g. . It is possible to eliminate one of the unknowns from (1), i.e. derive an equivalent (http://planetmath.org/Equivalent3) pair of polynomial equations
First we form the polynomial
the degree of which is less than . When is a solution of (1), then it satisfies
On the other hand, when is a solution of (3), then (2) implies that it satisfies also (1), except possibly in the case .
We can continue similarly until we arrive at a pair of equations
Substituting the roots of the former of the equations (4) into the latter one, which in practice is usually of first degree with respect to , one can get the corresponding values of . Hence one obtains all solutions of the original system of equations (1). Since the cases may yield wrong solutions, one should check them by substituting into (1).
Note. One can derive from the equations (1) an equation of lower degree also by eliminating from them the constant terms; the terms of resulting equation have as common factor or its higher power, which is removed by dividing.
|Title||elimination of unknown|
|Date of creation||2013-03-22 19:20:27|
|Last modified on||2013-03-22 19:20:27|
|Last modified by||pahio (2872)|
|Synonym||elementary method of elimination|