Question 1124074
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The sum of three numbers is 10.  The sum of twice the first&#8203; number, 5 times the second&#8203; 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&#8203; number:
second&#8203; number: 
third&#8203; number:
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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)
                                <font color = red>First number</font>, or {{{F = 11/11 = highlight(1)}}}

                                                  7(1) - S = 4 ---- Substituting 1 for F in eq (ii)
                                                       7 - S = 4
                                                          - S = 4 - 7
                                                          - S = - 3
                          <font color = red>Second number</font>, or {{{S = (- 3)/(- 1) = highlight(3)}}}

<font color = red>Third number</font>: 10 - F - S = 10 - 1 - 3 = 10 - 4 = <font color = red>6</font></font></font></font></b></pre>