Let P + (n) denote the largest prime of the integer n. Using the nilpotent Hardy-Littlewood method developed by Green and Tao, we give an asymptotic formula for where (F 1 , . . . , F t ) is a system of affine-linear forms of Z[X 1 , . . . , X d ] no two of which are affinely related and K is a convex body. This improves upon Balog, Blomer, Dartyge and Tenenbaum's work [1] in the case of product of linear forms.