Find all solutions $x,$ if they exist, to the system of congruences. If solution does not exist, prove the same.

$3+3+2+2+4+6=20$

$1$

$($a$)$

$x-=3(mod5)$

$x-=2(mod7)$

$x-=5(mod11)$

$($b$)$

$x-=5(mod6)$

$x-=2(mod35)$

$x-=47(mod143)$

$($c$)$

$11x-=33(mod55)$

$5x-=10(mod35)$

$7x-=35(mod77)$

$($d$)$

$x-=5(mod6)$

$2x-=6(mod8)$

$($e$)$

$2x+3y-=0(mod7)$

$5x+4y-=2(mod11)$

$($f$)$

$x+2y+3z-=2(mod3)$

$2x+3y+z-=1(mod5)$

$3x+y+2z-=4(mod7)$

$(2)$