Let the system of equations aligned & x+5 y-z=1 & 4 x+3 y-3 z=7 & 24 x+y+ z= aligned , R , have infinitely many solutions. Then the number of the solutions of this system, If x, y, z are integers and satisfy 7 x+y+z 77 , is

Step-by-step solution

For infinitely many solution aligned & =0 & | array ccc 1 & 5 & -1 4 & 3 & -3 24 & 1 & array |=0 & 1(3 +3)-5(4 +72)-1(4-72)=0 & -17 +3-4 72-4=0 & 17 =-289 aligned aligned & 1=0 & | array ccc 1 & 5 & -1 7 & 3 & -3 & 1 & -17 array |=0 & 1(-51+3)-5(-119+3 )-1(7-3 )=0 & -48+595-15 -7+3 =0 & 12 =540 aligned aligned & x+5 y-z=1 & 4 x+3 y-3 z=7 & 24 x+y-17 z=45 & Let z=1 & x+5 y=1+ ] 4 & 4 x+3 y=7+3 & 4 x+20 y=4+4 & -17 y=3- -17 aligned aligned & aligned y & = -3 17 , x =1+ - 5 -15 17 & = 32-12 17 7 & -3 17 + 32+12 17 + 7