Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every large odd integer N , and 5≤a≤8, a≤b, , the equation
is solvable with x being an almost-prime Pr(a,b) and the other variables primes, where r(a,b) is defined in the Theorem, in particular, r(6,7)=5. This result constitutes an refinement upon that of J. Brűdern.