Answer:
- 59 Natural number solutions
- 61 Whole number solutions
- infinitely many integer solutions
Explanation:
You want the number of integer solutions to the system of equations ...
- a + b + c = 1000
- a/5 + b/6 + c/7 = 160
Substitution
To ensure that a/5 and b/6 are integers, we can let a = 5n and b = 6m, where m and n are integers. Then the system of equations becomes a relation between m and n.
Substituting for 'a' and 'b' in the two equations, we have ...
5n +6m +c = 1000 ⇒ c = 1000 -5n -6m
(5n)/5 +(6m)/6 + c/7 = 160
Substituting the value of c in this second equation gives ...
n + m + (1000 -5n -6m)/7 = 160
7n +7m +1000 -5n -6m = 1120 . . . . . . multiply by 7
2n +m = 120 . . . . . . . . . . . . . . . . subtract 1000
Then we have ...
m = 120 -2n = 2(60 -n)
Solutions
If we want positive integer solutions, then n may take on values 1 .. 59. If we allow 0 as a solution value, then n may take on values 0 .. 60. If solutions may be positive or negative, then there are infinitely many solutions.
- 59 Natural number solutions
- 61 Whole number solutions
- infinitely many integer solutions
__
Additional comment
The solutions are ...
(a, b, c) = (5n, 12(60 -n), 7(40+n)) . . . . . for n a suitable integer
<95141404393>