We derive a method for solving N+m nonlinear algebraic equations in N+munknowns y≠R m and z≠R N of the form A(y)z+b(y)=0, where the(N+m) × N matrix A(y) and vector b(y) are continuously differentiable functions ofy alone. By exploiting properties of an orthonormal basis for null (AT(y)) the problem is reduced to solving m nonlinear equations in y only. These equations are solved by Newton's method inm variables. Details of computational implementation and results are provided.