A new method is proposed to solve systems of linear approximate equations Xθ&ap;y where the unknowns θ and the data y are positive and the matrix X consists of nonnegative elements. Writing the i-th near-equality X_i.θ/y_i&ap;1 the assumed model is X_i.θ/y_i=ζ_i with mutually independent positive errors ζ_i. The loss function is denned by Σw_i(ζ_i-1)logζ_i in which w_i is the importance weight for the i-th near-equality. A reparameterization reduces the method to unconstrained minimization of a smooth strictly convex function implying the unique existence of positive solution and the applicability of Newton's method that converges quadratically. The solution stability is controlled by weighting prior guesses of the unknowns θ. The method matches the maximum likelihood estimation if all weights wi are equal and ζ_i independently follow the probability density function ∝ t^<ω(1-t)>,0<ω.