The formula is called the Gaussian form of the thin-lens formula. another form of this formula, the Newtonian form, is obtained by considering the distance from the object to the first focal point and the distance from the second focal point to the image. I am going to show that is the Newtonian form of the thin-lens formula.
In this situation, my diagram is going to look as follows below:
Starting with the thin lens formula, , I note that the object distance is , and the image distance is , where the focal distances for the two lenses are equivalent. After making these substitutions into the thin lens formula, and solving for I obtain the equation . Now I can multiply each side of the equation by and solve for to obtain the final solution , which is the Newtonian form of the thin lens equation.