In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km?
Set $ \dfrac{1}{V} = 0$ and solve for $\dfrac{1}{[S]}$: 

$ 0 = \dfrac{K_m}{V_{max}}\dfrac{1}{[S]}+ \dfrac{1}{V_{max}} $

$ -\dfrac{1}{V_{max}} = \dfrac{K_m}{V_{max}}\dfrac{1}{[S]}$

$ -1 = {K_m}\dfrac{1}{[S]}$

$ -\dfrac{1}{K_m} = \dfrac{1}{[S]} = $ x-intercept