I have the following two systems of equations:
$a = x_{11} - \{x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}d\}$
$b = x_{12} - \{x_{12} + \frac{4 - \sqrt{2}}{7}a + \frac{2\sqrt{2} - 1}{7}c + \frac{4 - \sqrt{2}}{7}d\}$
$c = x_{21} - \{x_{21} + \frac{4 - \sqrt{2}}{7}a + \frac{2\sqrt{2} - 1}{7}b + \frac{4 - \sqrt{2}}{7}d\}$
$d = x_{22} - \{x_{22} + \frac{2\sqrt{2} - 1}{7}a + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c\}$
and
$a = x_{11} - [x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}d]$
$b = x_{12} - [x_{12} + \frac{4 - \sqrt{2}}{7}a + \frac{2\sqrt{2} - 1}{7}c + \frac{4 - \sqrt{2}}{7}d]$
$c = x_{21} - [x_{21} + \frac{4 - \sqrt{2}}{7}a + \frac{2\sqrt{2} - 1}{7}b + \frac{4 - \sqrt{2}}{7}d]$
$d = x_{22} - [x_{22} + \frac{2\sqrt{2} - 1}{7}a + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c]$
where $x_{ij}$ are known input variables, they are all in the [0; 1] closed interval, $\{\}$ denotes the decimal/fractional part, [] rounds to the closest integer and I'd like to solve this system for $a, b, c$ and $d$.
How do you usually start solving systems like this?
UPDATE:
I've substituted $d$ in the three other equations but then in the third one I have $c$ on both sides, the RHS having it as an argument for a double rounding:
$a = x_{11} - [x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}(x_{22} - [x_{22} + \frac{2\sqrt{2} - 1}{7}a + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c])]$
$b = x_{12} - [x_{12} + \frac{4 - \sqrt{2}}{7}a + \frac{2\sqrt{2} - 1}{7}c + \frac{4 - \sqrt{2}}{7}(x_{22} - [x_{22} + \frac{2\sqrt{2} - 1}{7}a + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c])]$
$c = x_{21} - [x_{21} + \frac{4 - \sqrt{2}}{7}a + \frac{2\sqrt{2} - 1}{7}b + \frac{4 - \sqrt{2}}{7}(x_{22} - [x_{22} + \frac{2\sqrt{2} - 1}{7}a + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c])]$
I can see I can't place the factors inside the round functions as the closest integer multiplied by the factor may be higher than the closest integer to (the expression itself multiplied by it). Shall I split the solution to as many cases as necessary, placing constraints on the sums different variables, forcing them inside intervals and then finding intersections as solutions?