Back-propagation(BP) algorithm needs a lot of time to train the artificial neural network (ANN) to get high accuracy level in classification tasks. So there have been extensive researches to process back-propagation algorithm on parallel processors. This paper prsents a linear systolic array which calculates forward-backward propagation of BP algorithm at the same time using effective space-time transformation and PE structure. First, we analyze data flow of forwared and backward propagations and then, represent the BP algorithm into data dapendency graph (DG) which shows parallelism inherent in the BP algorithm. Next, apply space-time transformation on the DG of ANN is turn with orthogonal direction projection. By doing so, we can get a snakelike systolic array. Also we calculate the interval of input for parallel processing, calculate the indices to make the right datas be used at the right PE when forward and bvackward propagations are processed in the same PE. And then verify the correctness of output when forward and backward propagations are executed at the same time. By doing so, the proposed system maximizes parallelism of BP algorithm, minimizes th enumber of PEs. And it reduces the execution time by 2 times through making idle PEs participate in forward-backward propagation at the same time.