19 and 42 are coprime, so we can use the CRT right away. Start with
[tex]x=19+42[/tex]
Taken mod 42, we're left with a remainder of 19. Notice that
[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]
so we need to multiply the first term by 3 to get the remainder we want.
[tex]x=19\cdot3+42[/tex]
Next, taken mod 19, we're left with a remainder of 4. Notice that
[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]
so we need to multiply the second term by 6.
Then by the CRT, we have
[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]
so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.
