The Michaelis–Menten relationship can be expressed in several algebraically equivalent linear forms used for plotting. Which statement correctly summarizes these alternative forms?

Introduction / Context:
Although the canonical Michaelis–Menten equation v = (Vmax * [S]) / (Km + [S]) is nonlinear, several linear rearrangements facilitate parameter estimation and diagnostic plotting. Three classic forms are Lineweaver–Burk, Hanes–Woolf, and Eadie–Hofstee.

Given Data / Assumptions:

• Symbols: r (or v) for rate, Cs (or [S]) for substrate, rmax (or Vmax) for maximum rate.
• Steady-state, initial-rate conditions apply.
• Algebraic rearrangements preserve equivalence if performed correctly.

Concept / Approach:
Demonstrate each rearrangement starting from r = (rmax * Cs) / (Km + Cs) and show equivalence to one of the standard plotting lines.

Step-by-Step Solution:

Verification / Alternative check:
Fitting the same data with these transformations yields identical parameter values in the absence of error; differences in practice reflect error-weighting, not algebraic inequivalence.

Why Other Options Are Wrong:

• Each individual linear form is valid; therefore “None of these” is incorrect.

Common Pitfalls:
Confusing sign conventions or mixing symbols; ensure r corresponds to v and Cs to [S]. Also note that linearizations can distort error structure; modern practice often fits the untransformed nonlinear model.

Final Answer:
All of these