SOLUTION: The sum of three numbers is 10. The sum of twice the first​ number, 4 times the second​ number, and 5 times the third number is 33. The difference between 6 times the f

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Question 1124158: The sum of three numbers is 10. The sum of twice the first​ number, 4 times the second​ number, and 5 times the third number is 33. The difference between 6 times the first number and the second number is 19. Find the three numbers.
first​ number:

second​ number:

third​ number:


**** No decimal number it has to be a whole number for the answer*****
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

let’s three numbers be x, y, and z

if the sum of three numbers is 10, we have
x%2By%2Bz=10........eq.1


if the sum of twice the first​ number, 4 times the second​ number, and 5 times the third number is 33, we have
2x%2B4y%2B5z=33........eq.2


if the difference between 6 times the first number and the second number is 19, we have
6x-y=19..............eq.3 ....solve for y
y=6x-19..........eq.a


go to 2x%2B4y%2B5z=33........eq.2, substitute y
2x%2B4%286x-19%29%2B5z=33
2x%2B24x-76%2B5z=33
26x%2B5z=33%2B76
26x%2B5z=109....solve for z
5z=109-26x
z=%28109-26x%29%2F5.........eq.b


go to x%2By%2Bz=10........eq.1 substitute y and z

x%2B6x-19%2B%28109-26x%29%2F5=10....solve for x

5x%2B30x-19%2A5%2B%28109-26x%29=10%2A5
5x%2B30x-95%2B109-26x=50
35x-26x=50%2B95-109
9x=145-109
x=36%2F9
highlight%28x=4%29


go to y=6x-19..........eq.a , plug in 4 for x
y=6%2A4-19
y=24-19
highlight%28y=5%29


go to z=%28109-26x%29%2F5.........eq.b , plug in 4 for x


z=%28109-26%2A4%29%2F5
z=%28109-104%29%2F5
z=5%2F5
highlight%28z=1%29

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The sum of three numbers is 10. The sum of twice the first​ number, 4 times the second​ number, and 5 times the third number is 33. The difference between 6 times the first number and the second number is 19. Find the three numbers.
first​ number:

second​ number:

third​ number:


**** No decimal number it has to be a whole number for the answer***** 
Let the 1st, 2nd and 3rd numbers be F, S, and T, respectively

Then: F + S + T = 10 --------- eq (i)

Also, 2F + 4S + 5t = 33 ------ eq (ii)

And, 6F - S = 19____S = 6F - 19 ------- eq (iii)

F + 6F - 19 + T = 10_____7F + T = 29______T = 29 - 7F -------- Substituting 6F - 19 for S in eq (i) ------- eq (iv)

2F + 4(6F - 19) + 5T = 33_____2F + 24F - 76 + 5T = 33_____26F + 5T = 109 -------- Substituting 6F - 19 for S in eq (ii) ------- eq (v)

26F + 5(29 - 7F) = 109 ------- Substituting 29 - 7F for T in eq (v)

26F + 145 - 35F = 109

- 9F = - 36

F, or 1st number = highlight_green%28matrix%281%2C3%2C+%28-+36%29%2F%28-+9%29%2C+%22=%22%2C+4%29%29 

S = 6(4) - 19 --------- Substituting 4 for F in eq ((iv)

S, or 2nd number = highlight_green%28matrix%281%2C3%2C+24+-+19%2C+%22=%22%2C+5%29%29

T = 29 - 7(4) ------ Substituting 4 for F in eq (iv)

T, or 3rd number = highlight_green%28matrix%281%2C3%2C+29+-+28%2C+%22=%22%2C+1%29%29