SOLUTION: The sum of three numbers is 10. The sum of twice the first​ number, 5 times the second​ number, and 6 times the third number is 53. The difference between 7 times the

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: The sum of three numbers is 10. The sum of twice the first​ number, 5 times the second​ number, and 6 times the third number is 53. The difference between 7 times the      Log On


   

Question 1124074: The sum of three numbers is 10. The sum of twice the first​ number, 5 times the second​ number, and 6 times the third number is 53. The difference between 7 times the first number and the second number is 4.
Find the three numbers.
first​ number:
second​ number:
third​ number:
Found 3 solutions by MathLover1, MathTherapy, josgarithmetic:
Answer by MathLover1(20855) 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, 5 times the second number, and 6 times the third number is 53, we have

2x%2B5y%2B6z=53......eq.2

if the difference between 7 times the first number and the second number is 4,we have
7x-y=4........eq.3

solve eq.3 for y

y=7x-4.....eq.a.....substitute in eq.2

2x%2B5%287x-4%29%2B6z=53......eq.2
2x%2B35x-20%2B6z=53
37x%2B6z=53%2B20..........solve for z
6z=73-37x
z=73%2F6-37x%2F6......eq.b
substitute y and z from eq.a and eq.b in eq.1

x%2B7x-4%2B73%2F6-37x%2F6=10......eq.1
8x-37x%2F6=10%2B4-73%2F6
8x-37x%2F6=14-73%2F6...both sides multiply by 6
48x-37x=84-73
11x=11
highlight%28x=1%29

go to y=7x-4.....eq.a substitute x
y=7%2A1-4
highlight%28y=3%29

go to z=73%2F6-37x%2F6......eq.b substitute x
z=73%2F6-%2837%2A1%29%2F6
z=36%2F6
highlight%28z=6%29



Answer by MathTherapy(10791) 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, 5 times the second​ number, and 6 times
the third number is 53.  The difference between 7 times the first number and the second number is 4.  

Find the three numbers.
first​ number:
second​ number: 
third​ number:
********************
This is MUCH simpler than the other person made it out to be.

Let the 1st number be F, and the 2nd, S
Since the sum of the 3 numbers is 10, the 3rd is: 10 - F - S

The sum of twice the first, 5 times the second, and 6 times the third is 53, gives us:
                          2F + 5S + 6(10 - F - S) = 53
                           2F + 5S + 60 - 6F - 6S = 53
                                                     - 4F - S = - 7 ---- eq (i)

The difference between 7 times the first and the second number is 4, gives us: 
                                                       7F - S = 4 ---- eq (ii)  

                                                    - 4F - S = - 7 --- eq (i)
                                                      7F - S =   4 ---- eq (ii) 
                                                         11F = 11 ---- Subtracting eq (i) from eq (ii)
                                First number, or F+=+11%2F11+=+highlight%281%29

                                                  7(1) - S = 4 ---- Substituting 1 for F in eq (ii)
                                                       7 - S = 4
                                                          - S = 4 - 7
                                                          - S = - 3
                          Second number, or S+=+%28-+3%29%2F%28-+1%29+=+highlight%283%29

Third number: 10 - F - S = 10 - 1 - 3 = 10 - 4 = 6

Answer by josgarithmetic(39780) About Me  (Show Source):
You can put this solution on YOUR website!
system%28x%2By%2Bz=10%2C2x%2B5y%2B6z=53%2C7x-y=4%29

What if first try to eliminate z?
system%286x%2B6y%2B6z=60%2C2x%2B5y%2B6z=53%29
Subtract later from former.
4x%2By=7

Revising the system now having just two variables,
syste%284x%2By=7%2C7x-y=4%29

Add the parts.
11x%2B0=11
highlight%28x=1%29

Using E3 of original system
7x-4=y
7-4=y
highlight%28y=3%29

Using E1 of original system
1%2B3%2Bz=10
highlight%28z=6%29