Find the $\gcd(81,237)$ and express it as a linear combination of $81$ and $237.$

Question

How are they finding the encircled part. I am trying my very best to understand it, but in vain.

Answer 1

$3 = 75 - (81 - 75 \cdot 1) \cdot 12$

$3 = 75 - (81 - 75) \cdot 12$ just multiplying by $1$

$3 = 75 - (12 \cdot 81 - 12 \cdot 75)$ from distributing the multiplication by $12$

$3 = 75 - 12 \cdot 81 + 12 \cdot 75$ from distributing the minus sign

$3 = 75 + 12 \cdot 75 - 12 \cdot 81$ from commutativity of $+$

$3 = 13 \cdot 75 - 12 \cdot 81$ from collecting the $75$s

Answer 2

Using distributivity : $75-(81-75)\times 12 =75-81\times 12 +75\times 12=$

using commutativity of sum: $=75\times 1 +75\times 12-81\times 12=$

usig distributivity : $=75\times(1+12)-81\times 12= 75\times 13 -81\times 12$