I have a system of equations shown as below:
$x_1x_3+x_5 = a$
$x_1x_4+x_5 =b$
$x_2 x_3 +x_5 =c$
$x_2x_4 +x_5 =d$,
where $x_1,x_2,x_3,x_4,x_5$ are variables and $a,b,c,d$ are constant.
Given any $(a,b,c,d)$, is it always possible to find a valid solution?
For example, given $a=1,b=c=d=0$, I can find a solution as $x_1=x_3=1,x_2=x_4=x_5=0$.
This is not a homework problem that I'm trying to find an answer. I just don't know where to start to tackle this problem. Can someone provide some hints?
Edit:
Please check if this statement is right: If $b-a = d-c \neq 0$ and ($a\neq c$ or $b\neq d$), then there is no solution; otherwise, there is always a solution.
No, it is not always possible. The solution is given, for arbitrary $x_4\neq 0$ by $$ x_1=\frac{ab - ad - b^2 + bd}{x_4(a - b - c + d)}, $$
$$ x_2=\frac{bc - bd - cd + d^2}{x_4(a - b - c + d)}, $$
$$ x_3=\frac{x_4(a - c)}{b - d}, $$
$$ x_5=\frac{ad - bc}{a - b - c + d}, $$
provided not one of the denominators equals zero. If $$ a-b-c+d=0, $$ then there is not always a solution. Take $(a,b,c,d)=(1,2,3,4)$. This can be found by using Buchberger's algorithm, or by substitution.