Let \(V\subseteq \mathbb {A}^n\) be an affine variety, with \(\mathcal {I}_a(V)=(f_1,\ldots , f_m)\) and let \(P=(p_1,\ldots , p_n)\) be a point of V. Let r be a line passing through P, so that r has parametric equations of the form $$ x_i=p_i+\lambda _it, \quad \text {with}\quad t\in \mathbb {K}\quad \text {for}\quad i=1,\ldots , n,\quad \text {and}\quad (\lambda _1,\ldots , \lambda _n)\ne \mathbf {0}. $$ The polynomial system in t $$ f_i(t):=f_i(p_1+\lambda _1t,\ldots ,p_n+\lambda _nt)=0, \quad i=1,\ldots , m $$ has the solution \(t=0\). If the polynomials \(f_1(t),\ldots , f_m(t)\) are all identically 0, this means that r is contained in V.