imc

Since $(1,2)$ is a solution to both equations, plugging in $x=1$ and $y=2$ will give the values of $a$ and $b$. $1=\frac{1}{4}\cdot 2+a$ $a=\frac{1}{2}$ $2=\frac{1}{4}\cdot 1+b$ $b=\frac{7}{4}$ So $a+b=\frac{1}{2}+\frac{7}{4}=\boxed{\frac{9}{4}}$.