Answer
No.
Work Step by Step
A is a 10x12 matrix with 3 free variables
By the Rank Theorem
$rank A + dim (Nul A) = n$
$rank A + 3 = 12$
$rank A = 9$
so
$Span (Col A)$ is $R^{9}$
Since solutions of the transformation are in $R^{10}$, $Span (Col A)$ must be 10 in order to account for every solution in $R^{10}$ so the scientist cannot be sure that if the right side is changed there will be a solution because $Span (Col A)$ is only 9
