Solve a system of linear equations with binary variables

[lopeslimagabriel]

I'm having trouble defining an algorithm that can solve a system of linear equations with binary variables.
Example given the following system of linear equations:

x_12 + x_18 + x_28 = 0
x_12 - x_18 - x_28 = 0
x_12 + x_13 + x_28 - x_38 = 0
-x_13 + x_14 - x_38 - x_48 = - 2
-x_14 + x_15 - x_48 - x_58 = -2
x_15 + x_16 + x_58 - x_68 = 0

Knowing that the values of x_12 = 0, x_18 = 0 and x_28 = 0, it is possible to create an algorithm that solves this system with the other variables assuming only values of 0 or 1.