How to generate a list (or even a sequence or a string) containing {x[1]==0&&x[2]==0&&…&&x[10]==0?

I need to solve a bunch of systems of n equations in n unknowns where n varies. I can do them, one by one, but I wish to automate the process.

The first equation equals 1 and the rest all equal zero. I currently type in each set of equations manually as follows. I have the n unknowns in a list: list1={r,s,…,t}. I have the expressions for the LHS in a 2nd list: list2={expr1, …, exprn]. I write sol=Solve[list2[[1]]==1&&list2[[2]]==0 … &&list2[[n]]==0,list1]

I would like to just type Solve[expr,list] where expr is something like list2[[1]]==1&&list2[[2]]==0 … &&list2[[n]]==0. I experimented with Table, RowBox, BoxData, and even CellPrint but I don’t know enough to write an expression involving && and == that will work inside the Solve function.