Suppose that we have a linear system of equations
$$Ax=b$$
where $A$ is a $3 \times 3$ matrix and $x$ and $b$ are $3$-vectors. Let $y$ denote the solution of this system of equations. I want to change matrix $A$ such that the new solution is vector $z$ in which
$$z_1 > y_1, \qquad z_2 = y_2, \qquad z_3 < y_3$$
Is there a systematic way to achieve this? In other words, I want a systematic way of finding out what changes I should introduce in matrix $A$ such that
some entries of the new solution $z$ are greater than the corresponding entries of the old solution $y$.
other entries of the new solution $z$ are equal to the corresponding entries of the old solution $y$.
some other entries of the new solution $z$ are less than the corresponding entries of the old solution $y$.
Is there a method or technique to achieve this? What is it called? Thank you.