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We present the results of a new study investigating the relationship between observed Ly$α$ equivalent width ($W_λ$(Ly$α$)) and the metallicity of the ionizing stellar population ($Z_{\star}$) for a sample of $768$ star-forming galaxies at $3 \leq z \leq 5$ drawn from the VANDELS survey. Dividing our sample into quartiles of rest-frame $W_λ$(Ly$α$) across the range $-58 \unicode{xC5} \lesssim$ $W_λ$(Ly$α$) $\lesssim 110 \unicode{xC5}$ we determine $Z_{\star}$ from full spectral fitting of composite far-ultraviolet (FUV) spectra and find a clear anti-correlation between $W_λ$(Ly$α$) and $Z_{\star}$. Our results indicate that $Z_{\star}$ decreases by a factor $\gtrsim 3$ between the lowest $W_λ$(Ly$α$) quartile ($\langle$$W_λ$(Ly$α$)$\rangle=-18\unicode{xC5}$) and the highest $W_λ$(Ly$α$) quartile ($\langle$$W_λ$(Ly$α$)$\rangle=24\unicode{xC5}$). Similarly, galaxies typically defined as Lyman Alpha Emitters (LAEs; $W_λ$(Ly$α$) $>20\unicode{xC5}$) are, on average, metal poor with respect to the non-LAE galaxy population ($W_λ$(Ly$α$) $\leq20\unicode{xC5}$) with $Z_{\star}$$_{\rm{non-LAE}}\gtrsim 2 \times$ $Z_{\star}$$_{\rm{LAE}}$. Finally, based on the best-fitting stellar models, we estimate that the increasing strength of the stellar ionizing spectrum towards lower $Z_{\star}$ is responsible for $\simeq 15-25\%$ of the observed variation in $W_λ$(Ly$α$) across our sample, with the remaining contribution ($\simeq 75-85\%$) being due to a decrease in the HI/dust covering fractions in low $Z_{\star}$ galaxies.