= Solution
A Gaussian <generalised additive model> with the identity link specifies
$$
\boxed{Y_i=\beta_0+\sum_{j=1}^p f_j(x_{ij})+\varepsilon_i,\qquad\varepsilon_i\text{ independent }N(0,\sigma^2).}
$$
The $f_j$ are unknown smooth functions estimated from the data, rather than necessarily straight lines. The model is additive in their effects, so it does not automatically contain <interactions> between <covariates>. To identify the intercept separately from the smooths, impose a constraint such as $\sum_i f_j(x_{ij})=0$ for each $j$. Linear terms can also be included alongside smooth terms. More generally a specified <link function> can make $g(E(Y_i))$ additive; for the normal identity-link model the mean itself has the displayed form. Smoothness is typically controlled through a basis and a <roughness penalty>, preventing an unrestricted smooth from interpolating all the noise.
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