A new method for solving non-Hermitian and normal positive-definite systems of linear equations is received. Theoretical analysis shows that if \(\sigma_{\max}\leq \lambda_{\min}\) the method converges to the unique solution of the system of linear equations \(Ax=b\), where \(A\in \mathbb{C}^{n\times n}\) is a large sparse non-Hermitian positive-definite matrix and \(x, b \in \mathbb{C}^n\) for any positive \(\alpha\). A bound for the spectral radius of the iteration matrix is found. An inexact version of the method is also introduced. When the Hermitian part of the coefficient matrix is dominant, the new method performs very well; however, as the skew-Hermitian part becomes dominant, the Hermitian/skew-Hermitian splitting method is better.