There are many posts here and elsewhere asking for algorithms to simplify simple arithmetic expressions. For example, this question asks how to simplify the expression $axc + byc + ayc + bxc$. Symbolics and xcas are two examples among many of Computer Algebra Systems (CAS) that can do this. But Symbolics and xcas perform differently: Symbolics can simplify the expression above and turn it into $cx(a + b) + cy(a + b)$, whereas xcas completely factorizes the expression into $(a + b)c(x + y)$.
I was wondering if there is any CAS that simplifies expressions based on some set of equations that you can input into the system? As an example, assume that we knew $a + b = 2$ and $x + y = -1$: is there any known (implemented) algorithm that given said equations, would factorize the expression above into simply $-2c$ ? Perhaps it's more pertinent to ask: could such an algorithm exist?
In the example above, we can set $y = -1 - x$ and $b = 2 - a$ (as pointed out in a comment), but in more complicated examples it might not be possible.