# In the Lineweaver-Burk Plot, why does the x-intercept = -1/Km?

Taking the reciprocal of both sides of the Michaelis-Menten equation yields the Lineweaver-Burk Equation:

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

Plotting a $\dfrac{1}{V}$ vs. $\dfrac{1}{[S]}$ graph, I am told that:

y-int $= \dfrac{1}{V_{max}}$ and

x-int $= -\dfrac{1}{K_m}$

How are these relationships derived from the lineweaver-burk plot? I can see how the y-intercept can be equal to $\dfrac{1}{V_{max}}$ if $\dfrac{1}{[S]} = 0$, but I don't see how x-int $= -\dfrac{1}{K_m}$ by setting $\dfrac{1}{V} = 0$? Can someone demonstrate how these relationships were derived?

• Equation of a line: y=mx+c or y=m(x-d) where c is y-intercept and d is x-intercept. The general equation of a line is y-y₁=m(x-x₁) – WYSIWYG Oct 24 '16 at 7:03

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