Given a pair of linear congruences of the form below where the $m_i$ are not necessarily coprime. $$a_1 x \equiv b_1 \pmod{m_1}$$ $$a_2 x \equiv b_2 \pmod{m_2}$$
I know that if the $a_1 = a_2 = 1$, then we can check if congruences are compatible by checking if, $$ a_1 = a_2 \pmod{gcd(m_2, m_2)}$$
However, I am wondering if there is a way to check compatibility of two congruences without first finding the inverses of $a_1$ and $a_2$ to simplify the system to the form of,
$$x \equiv a_1^{-1} b_1 \pmod{m_1}$$ $$x \equiv a_2^{-1} b_2 \pmod{m_2}$$
I am hoping to avoid the computational cost of computing an inverse if the equations are not compatible.
