Let \(A\) be a finitely generated algebra over a field \(k\) of characteristic \(0\), and fix a finite set \(a_1,\dots,a_n\) of generators. For each \(\alpha=(\alpha_1,\dots,\alpha_n)\in P\), the additive monoid of all vectors of integer non-negative components, we have the associated standard monomial \(a^\alpha=a_1^{\alpha_1}\cdots a_n^{\alpha_n}\). Assume that \(A\) is a solvable polynomial algebra, that is, the set \(\{a^\alpha\mid\alpha\in P\}\) is a \(k\)-basis for \(A\), and that \(a^\alpha a^\beta=c_{\alpha,\beta}a^\beta a^\alpha+p_{\alpha,\beta}\) for every \(\alpha,\beta\in P\), where the leading monomial of \(p_{\alpha,\beta}\in A\) is strictly less that \(\alpha+\beta\) with respect to a fixed monomial ordering \(\leq\) on \(P\), and the \(c_{\alpha,\beta}\)'s are nonzero scalars in \(k\). Accordingly with \textit{A. Kandri-Rody} and \textit{V. Weispfenning} [J. Symb. Comput. 9, No. 1, 1-26 (1990; Zbl 0715.16010)], basic algorithms for computation of Gröbner bases can be transferred from commutative polynomial algebras to \(A\) in a satisfactory way. When the tails \(p_{\epsilon_i,\epsilon_j}\) are linear polynomials in \(a_1,\dots,a_n\), where \(\epsilon_i\) denotes the \(i\)-th vector in the canonical basis, and the monomial ordering is the graded lexicographical one, then \(A\) is said to be a linear solvable polynomial algebra. In this case, \textit{H. Li} observed [in Commun. Algebra 27, No. 5, 2375-2392 (1999; Zbl 0937.16035)] that the aforementioned Gröbner basis effective methods can be used to give an algorithm for the computation of the Gelfand-Kirillov dimension of cyclic left \(A\)-modules. This observation was done a little bit earlier by \textit{J. L. Bueso, F. J. Castro, J. Gómez Torrecillas} and \textit{F. J. Lobillo} [in Lect. Notes Pure Appl. Math. 197, 55-83 (1998; Zbl 0898.17007)], and, more recently [in Lect. Notes Pure Appl. Math. 221, 33-57 (2001; Zbl 0985.16015)], where such an algorithm was developed for arbitrary solvable polynomial algebras. Based on this Gelfand-Kirillov dimension effective computation, the authors give an interesting Elimination Lemma in the linear case, extending the one given by \textit{D. Zeilberger} [in J. Comput. Appl. Math. 32, No. 3, 321-368 (1990; Zbl 0738.33001)] for holonomic modules over the Weyl algebra \(A_n(k)\). As the authors nicely explain, this result opens a way to the solution of the extension/contraction problem stemming from the automatic proving of multivariate identities with respect to the \(\partial\)-finiteness in the sense of \textit{F. Chyzak} and \textit{B. Salvy} [in J. Symb. Comput. 26, No. 2, 187-227 (1998; Zbl 0944.05006)]. This Elimination Lemma is deduced from the main new result in the paper (Theorem 3.10), which reads: if \(L\) is a left ideal in a linear solvable polynomial algebra \(A\), then the Gelfand-Kirillov dimension of \(A/L\) is the maximum of the cardinals of the subsets \(\{a_{i_1},\dots,a_{i_r}\mid i_1<\cdots