Off-diagonal Rado number for $x+y+c=z$ and $x+qy=z$

Ramsey-type problems for linear equations began with Schur’s theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations $({\mathcal{E}}_1,{\mathcal{E}}_2)$ and determines the minimum integer $N$ for which any red-blue coloring of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},N\}$ forces either a red solution of the equation ${\mathcal{E}}_1$ or a blue solution of the equation ${\mathcal{E}}_2$. In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms $x+y+c=z$ and $x+qy=z$. We determine the exact two-color off-diagonal Rado number $R_2(c,q)$ associated with this system of equations.