In most cases no, the answer is yes iff $y_i=y_j\forall i,j\in\{1,2,\cdots,n\}$
Consider $x_1 + x_2 + \dots+ x_n = A$ and that the condition I stated are false.
In this case I can rewrite the $y_1x_1 +y_n x_2 + \dots+ y_nx_n=B$ it as $w_1u_1+w_{2}u_{2}+\cdots+w_\ell u_\ell=B$ where $1<\ell\le n$, $w_k=y_h$ for some $h$ and $u_k$ is combination of some values of $x_s$.
This is equation is the most reduce form, i.e. has the least amount of variables we can get, so unless I have only one variable I can't solve it, but one of my conditions is that not all of $y_i$ are equal, so I have at least 3 variables thus I can't solve this.