Can a system with a response Y(x) = ax + c be called linear?
If two systems, A and B, are linear then for a cascade of the two systems, the order does not matter. That is, AB = BA:
For example, let system A be an ideal gain of 10 stage while system B is an ideal 1st order low-pass filter with unity DC gain.
Since both stages are linear, the cascade of the two systems is a low-pass filter with a DC gain of 10 regardless of whether B follows A or A follows B in the cascade.
Now, see that a system with gain and offset is not a linear system. For example, let system A be as before but system B is now unity gain with a constant offset of 1.
For the cascade AB, the output is the input scaled by 10 plus an offset of 1.
However, for the cascade BA, the output is the input scaled by 10 plus an offset of 10 and so system B is not a linear system.
Another definition of linearity is the following: if \$y_1\$ is the output of a system given input \$x_1\$ and \$y_2\$ is the output of the same system given input \$x_2\$, then given the input \$x_3 = a_1x_1 + a_2x_2\$, the output is \$y_3 = a_1y_1 + a_2y_2\$ if and only if the system is linear.
For the case of system B with unity gain and offset 1, we have
$$y_1 = x_1 + 1$$
$$y_2 = x_2 + 1$$
$$y_3 = a_1x_1 + a_2x_2 + 1 \ne a_1y_1 + a_2y_2 = a_1x_1 + a_2x_2 + a_1 + a_2$$
Thus, system B is therefore not a linear system and so the answer to the quoted question is no.