Linear equation with form c'Ax Suppose $A$ is a given $n\times n$ matrix, $c$ a given $n \times 1$ vector, and $x$ an unknown $n\times 1$ vector. Given $c'x$, is it possible to find $c'Ax$? Conversely, given $c'Ax$, is it possible to find $c'x$?

In general it is impossible to find $x$ given $c'x = y$: you can see it as one equation with $n$ unknowns, which has $\infty^{n-1}$ solutions. All these are parametrized by the $n-1$ values $t_1,\ldots, t_{n-1}$ and are given by (for example) $$\begin{cases} x_1 = t_1,\\\ \vdots \\\ x_{n-1} = t_{n-1}, \\\ x_n = \frac{y}{c_n} - \frac{c_1}{c_n} t_1 -\ldots - \frac{c_{n-1}}{c_n} t_{n-1}.\end{cases}$$

Pick any one of these, and you can compute $c'Ax$. However, there is not only one solution, but infinitely many.

The same exact discourse holds for the reverse statement: let $b' = c'A$, you can find all the $x$ such that $b'x = y$ and then compute $c'x$ for each one of these.