First, I believe there is a trivial error. The second equation should have another $\Delta t$ multiplying everything on the right. It is divided out later when the equation I set equal to 0.
Given that $L$ is a function of $ x_7, x_8, x_9$ how can he justify evaluating it at single points?
Also, why are points 8 and 9 used rather than 7 and 8?
For context, this is the last equation on page 112 and the first equation on page 113 of Susskind's The Theoretical Minimum.