smc

$4a+7=7b+4⟹7b=4a+3⟹b=\frac{4a+3}{7}$. $4a+3\equiv 0⟹4a\equiv 4⟹a\equiv 1(\mathrm{mod}7)$. The smallest possible value of $a$ is $8$. Then, $b=\frac{4\cdot 8+3}{7}=\frac{35}{7}=5$. However, the digit $7$ is not valid in base $5$, so we have to try a larger value. $a=8+7=15$, gives a value of $\frac{15\ast 4+3}{7}=9$, for $b$, which is valid. $15+9=24$, which is $\mathrm{XXIV}$ as a roman numeral, and thus the answer is $\boxed{\mathrm{XXIV}}.$