How is dimension of $V(g_1|_{x=0})$ same as $V(g_1) \cap V(x_1)$?

Question

Suppose I have $g_1, \ldots, g_t \in \mathbb{C}[x_1, ..., x_n]$. Let $$ V_1=\{ \mathbf{x} \in \mathbb{C}^n : g_j(\mathbf{x}) = 0, 1 \leq j \leq t \} \cap \{ \mathbf{x} \in \mathbb{C}^n : x_1 = 0 \} $$ and $$ V_2 = \{ \mathbf{y} \in \mathbb{C}^{n-1} : g_j(0, \mathbf{y}) = 0, 1 \leq j \leq t \}. $$ These two algebraic sets are basically the same... but how does one go about to show that they have the same dimension? (as $V_1 \in \mathbb{A}^n$ and $V_2 \in \mathbb{A}^{n-1}$) Thank you very much.