Family Of Linear Equations

First, realize immediately that because $a_i$ is the y-intercept, then $a_i=ib_i$ .

Now, we are given that $b_i$ is the unique x-intercept. This means that $b_i$ is the solution to $0=a_ix+ib_i$ , or $0=a_ib_i+ib_i$ .

This can be factored into $b_i(a_i+i)=0$ . We know that $a_i=ib_i$ , so $b_i(ib_i+i)=0$ or $ib_i(b_i+1)=0$

This means that $b_i=0,-1$ . However, if $b_i=0$ , then $a_i=0$ and there wouldn't be a unique x-intercept. So $b_i=-1$ for all $i=1\to 2014$ .

Therefore, $a_i=-i$ for all $i=1\to 2014$ .

Therefore, $\displaystyle\sum_{i=1}^{2014}a_i+b_i=-\left(2014+\dfrac{2014\cdot 2015}{2}\right)=-2031119$ .

Our answer is therefore $-2031119\pmod{1000}=\boxed{881}$ and we are done.