= Solution
Use unambiguous symbols $\gamma_{SV},\gamma_{SL},\gamma_{LV}$ for solid–vapor, solid–liquid and liquid–vapor interfacial energies. The printed description repeats an interface subscript, so it cannot be used literally for two different interfaces.
For a smooth homogeneous equilibrium surface, take a horizontal displacement of the <contact line>. The substrate-parallel <force> balance is
$$
\gamma_{SV}-\gamma_{SL}-\gamma_{LV}\cos\theta_C=0.
$$
Thus \b[the contact-angle relation is <Young's equation>]:
$$
\boxed{\cos\theta_C=\frac{\gamma_{SV}-\gamma_{SL}}{\gamma_{LV}}.}
$$
The same result follows by varying interfacial free energy at fixed liquid volume. It assumes equilibrium, no contact-line pinning, and negligible line-tension and roughness corrections. If the inferred cosine lies outside $[-1,1]$, the finite-angle equilibrium construction fails and a wetting limit must be considered.
The name in the question should be distinguished from the <Young–Laplace equation>, which relates a pressure jump to interface curvature. The horizontal interfacial balance derived here is Young's equation; it does not require introducing a pressure-curvature formula.
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