It is shown that the joint maximum likelihood estimator of slope and intercept of the regression line in the classical (known error-variance ratio) linear structural relationship model can be represented as a solution of a two- dimensional M- equation. Therefore, it is possible to use a general saddlepoint approximation for multidimensional M- equations. Under normality assumptions we express the solution of the implicit multivariate “centering equation” in an explicit form. This allows a considerable saving of computing time. By integrating out numerically an unwanted variable one is also able to find the saddlepoint approximation for the slope- estimator. Numerical examples illustrate the efficiency of the approximation.