= Solution
Let $X=Y\cup Z$, where $Y$, $Z$, and $Y\cap Z$ are path-connected subcomplexes containing the basepoint; small open neighbourhoods may be used to reduce to the usual open-cover statement. The <Seifert-van Kampen theorem> says that
$$
\boxed{\pi_1(X)\cong
\pi_1(Y)*_{\pi_1(Y\cap Z)}\pi_1(Z).}
$$
More explicitly, this is the quotient of the <free product> $\pi_1(Y)*\pi_1(Z)$ by the normal closure of
$$
i_{Y*}(\gamma)i_{Z*}(\gamma)^{-1},
\qquad \gamma\in\pi_1(Y\cap Z).
$$
Equivalently, $\pi_1(X)$ is the pushout of the two inclusion-induced homomorphisms.
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