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The sun of three numbers is 19. The sum of twice the first number, 5 times the second number, and 6 times the third number is 77. The difference between the first number and the second number is 19. Find the three number.

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The three numbers are 56/5, -39/5 and 78/5.

Step-by-step explanation:

EQUATION 1:

First number: x

Second number: y

Third number: z

x + y + z = 19

EQUATION 2:

Sum of following is 77

Twice the first number: 2x

5 times the second number: 5y

6 times the third number: 6z

So, 2x + 5y + 6z = 77

EQUATION 3:

Difference between first and second number is 19.

x - y = 19

Equation 1: x + y + z = 19

Equation 2: 2x + 5y + 6z = 77

Equation 3: x - y = 19

1. Find x in terms of y

 x - y = 19

 x = 19 + y

2. Find y in terms of z by putting the value of x in first and second equation

   x + y + z = 19

   (19 + y)+ y + z = 19

   2y + z = 19 - 19

   2y + z = 0

   y = -z/2

and

2(19+y)+5y+6z=77

now putting the value of y in this equation

2(19-z/2)+5(-z/2)+6z=77

38 - z -5z/2 +6z = 77

5z/2 + 38 = 77

5z/2 = 39

z = 78/5

Now, y = -z/2

y = (-78/5)/2

y = -39/5

and  x = 19 + y

x = 19 - 39/5

x = 56/5

Therefore, the three numbers are 56/5, -39/5 and 78/5.

Keyword: Sum

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