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The Rees algebra R[It] plays an important role in commutative algebra because it encodes the asymptotic behavior of the ideal I. An important problem in the theory of Rees algebras is to determine defining equations of the Rees algebras. More generally, given any ideals I1, . . . , Ir in a ring R, one would like to describe defining equations of the multi-Rees algebra R[I1t1, . . . , Irtr]. This is a problem in pure algebra but when we deal with ideals in a polynomial ring we may use computational methods to find the equations. In this talk I describe defining equations of the multi-Rees algebras of certain family of ideals using results in pure and computational commutative algebra. To do this I start the talk by discussing some concepts in computational commutative algebra.