Why Did Moses Slay The Egyptian?
To get an understanding about the subject, lets look at some Scriptures.
“And it came to pass in those days, when Moses was grown, that he went out unto his brethren, and looked on their burdens: and he spied an Egyptian smiting an Hebrew, one of his brethren.  And he looked this way and that way, and when he saw that there was no man, he slew the Egyptian, and hid him in the sand.  And when he went out the second day, behold, two men of the Hebrews strove together: and he said to him that did the wrong, wherefore smitest thou thy fellow?  And he said, who made thee a prince and a judge over us? intendest thou to kill me, as thou killedst the Egyptian? And Moses feared, and said, surely this thing is known.  Now when Pharaoh heard this thing, he sought to slay Moses. But Moses fled from the face of Pharaoh, and dwelt in the land of Midian: and he sat down by a well.”
The purpose of this study is to understand why Moses killed the Egyptian.
In the writings of Josephus in The Antiquities of the Jews in Book 2, Chapter 9, Section 3:
“A man whose name was Amram, one of the nobler sort of the Hebrews, was afraid for his whole nation, lest it should fail, by the want of young men to be brought up hereafter, and was very uneasy at it, his wife being then with child, and he knew not what to do. Hereupon he betook himself to prayer to God; and entreated him to have compassion on those men who had nowise transgressed the laws of his worship, and to afford them deliverance from the miseries they at that time endured, and to render abortive their enemies’ hopes of the destruction of their nation. Accordingly God had mercy on him, and was moved by his supplication. He stood by him in his sleep, and exhorted him not to despair of his future favors. He said further, that he did not forget their piety towards him, and would always reward them for it, as he had formerly granted his favor to their forefathers, and made them increase from a few to so great a multitude. He put him in mind, that when Abraham was come alone out of Mesopotamia into Canaan, he had been made happy, not only in other respects, but that when his wife was at first barren, she was afterwards by him enabled to conceive seed, and bare him sons. That he left to Ismael and to his posterity the country of Arabia; as also to his sons by Ketura, Troglodytis; and to Isaac, Canaan. That by my assistance, said he, he did great exploits in war, which, unless you be yourselves impious, you must still remember. As for Jacob, he became well known to strangers also, by the greatness of that prosperity in which he live