You "understanding" is not quite correct. If a system of modulus equation has co-prime moduliandeach equation has a solution itself,thenthere exist values satifying all of them.

However, here, because 10 and 35 have common factor, 5, there is NO x such that 10x is one more than a multiple of 35 there is no x satisfying the last equation alone and so cannot be a value of x satisfying all of them.