Stuck on an affine cipher exercise
If I'm reading your question right, you've made two simple math mistakes:


The solution to $-1 \equiv -15a \pmod{26}$ isn't $a = 15$; it's $a \equiv 15^{-1} \pmod{26}$, or $a = 7$.
I'm not sure where you got $b = 3$, but it's not correct, either.  The right solution, using the correct value of $a = 7$, is $b = 19 - 2a = 19 - 14 = 5$.


(FWIW, using the incorrect value $a = 15$ should've given you either $b = 15$ or $b = 25$, depending on which of the two original congruences you plugged the wrong $a$ into.)

With the correct values of $a$ and $b$, your decryption formula works just fine: $a^{-1} = 7^{-1} \equiv 15 \pmod{26}$, so:

\begin{aligned}
15 \times (19 - 5) &amp;\equiv \phantom{0}2 \pmod{26} \\
15 \times (20 - 5) &amp;\equiv 17 \pmod{26}.
\end{aligned}

Both of your mistakes seem like simple lapses of concentration.  Maybe you just need to take a break from your homework and go get some sleep first.