dare to electricity

The polarities of the $6-V$ and $2-V$ voltage sources are opposite, therefore the resultant equivalent voltage source is a $4-V\space (6-2)$ source in place of the $6-V$ source, while the $2-V$ source is replaced with a short-circuit. The resultant equivalent circuit is a $4-V$ source in series with $R$ , $5-\Omega$ and $20-\Omega$ resistors.

Since the potential difference $V_{XY}$ between $X$ and $Y$ is $2.5V$ , then the current in the circuit is given by:

$\quad \quad I = \dfrac {V_{XY}}{5+20} = \dfrac {2.5}{25} = 0.1A$

The potential difference $V_R$ across $R$ is given by:

$\quad \quad V_R = 4 - V_{XY} = 4 - 2.5 = 1.5V$

$\quad \quad \Rightarrow R = \dfrac {V_R}{I} = \dfrac {1.5}{0.1} = \boxed {15}\Omega$