Finding the value of stoichiometric coefficient x
The relationship between $K_p$ and $K_c$ is 

$$K_p=K_c(RT)^{\Delta n}$$

where $\Delta n$ is the change in the number of moles of gas molecules over the course of the reaction. The only way you could know this change is from the stoichiometric coefficients. The $\Delta n$ term derives from the laws of mass action in which all partial pressures and/or concentrations are raised to the stoichiometric coefficients.

Consider:

$$\ce{2NO(g) + O2(g) &lt;=&gt; 2NO2(g)}$$
$$K_c = \dfrac{[\ce{NO2}]^2}{[\ce{NO}]^2[\ce{O2}]}$$
$$K_p = \dfrac{P^2_{\ce{NO2}}}{P^2_{\ce{NO}}P_{\ce{O2}}}$$
$$P_x = \dfrac{n_x}{V_x}RT=[\ce{X}]RT$$
$$K_p = \dfrac{([\ce{NO2}]RT)^2}{([\ce{NO}]RT)^2([\ce{O2}]RT)}$$
$$K_p = \dfrac{[\ce{NO2}]^2}{[\ce{NO}]^2[\ce{O2}]}\dfrac{(RT)^2}{(RT)^3}=K_c(RT)^{-1}$$
$$\Delta n = -1 =2-(2+1)$$