Affine transformations and letter frequencies

$4 * a + b \equiv 12 \mod{26}$ $19 * a + b \equiv 23 \mod 26$ $$

Solving the system we obtain: $$

$15 * a \equiv 11 \mod{26}$ $$

Using a repeated application of the Euclidean algorithm you can verify that $7$ is an inverse of $15$ modulo $26 \implies$ $a \equiv 77 \mod 26 \equiv 25 \mod{26} \implies b \equiv -88 \mod{26} \equiv -10 \mod{26} \equiv 16 \mod{26}.$ $$

$\implies 25 * P + 16 \equiv C \mod{26} \implies 25 * P \equiv (C - 16) \mod{26}$ . For $25 * \bar{a} \equiv 1 \mod{26} \implies \bar{a} \equiv -1 \mod{26} \equiv 25 \mod{26},$ so $25$ is an inverse of $25$ modulo $26 \implies$ $P \equiv 25 * (C - 16) \mod{26}.$

$$

$\therefore \boxed{\bar{a} + b = 41 \equiv 15 \mod{26}}.$