Answer:
The numbers are 8 , 5 , 1
Step-by-step explanation:
Let the three numbers be x,y,z
Sum of three numbere is 14
⇒ x +y + z = 14 ---------------(I)
The sum of twice the first number, 4 times the second number, and 5 times the third number is 41. 
⇒ 2x + 4y + 5z = 41 -----------(II)
The difference between 7 times the first number and the second number is 51.
⇒  7x -y = 51 ---------------(III)
Multiply equation (I) by (-5) and then add this wiht equation (II). Thus 'z' will be eliminated.
(I)*(-5)        -5x - 5y - 5z = -70
(II             <u>   2x + 4y + 5z = 41</u>   {Now add}
                 -3x   - y          = - 29  ------------(IV)
Multiply (III by (-1) and then add it with equation (IV)
(III)*(-1)        -7x + y = -51
(IV)            <u> -3x - y = -29 </u>   {Now add}
                   -10x     = -80
                           x = -80 ÷ (-10)
x = 8
Plugin x = 8 in equation (III) and we obtain the value of y
7*8 - y = 51
 56 - y = 51
      -y = 51 - 56
     -y = -5
    y = -5 ÷ (-1)
y = 5
Plugin x =8 and y =5 in equaiton (I)
8 + 5 + z = 14
     13 + z = 14
          z  = 14 - 13
  z = 1