I assume that you want to work with linear equations.

Let's say your equations 1 and 2 were of the form: $\sum_{i=1}^{n} a_{ij}.x_i \leq b_j for j\in\{1,2\}$.

The main idea to solve $z_j=1 \implies \sum_{i=1}^{n} a_{ij}.x_i \leq b_j$ is to find a big number $M_j$ which is an upper bound for $\sum_{i=1}^{n} a_{ij}.x_i - b_j$.

Thus

$$\sum_{i=1}^{n} a_{ij}.x_i \leq b_j + M_j(1-z_j)$$

Note that, if $z_j=0$ then $\sum_{i=1}^{n} a_{ij}.x_i \leq b_j +M_j$ is feasible for any $x$ and, if $z_j =1$, the constraint $\sum_{i=1}^{n} a_{ij}.x_i \leq b_j$ will cut some values for $x$.