Since a,b and c are all non-negative numbers, then all of them are ≥ 0

Since a + b + c = 30, then none of these numbers alone can be greater than 30.

(Note that the question did not specify that a ≠ b ≠ c, so it is possible to have all 3 identical, or 2 identical or all different numbers)

We can deduce that 0 ≤ a,b,c, ≤ 30

Next, we are given the following :

① a + b + c = 30 and ② 3a + b - c = 50


② - ① :

2a - 2c = 20
a - c = 10
c = a - 10 ③

① + ② :

4a + 2b = 80
2a + b = 40
b = 40 - 2a ④


We are also given N = 5a + 4b + 2c

Sub ③ and ④,

N = 5a + 4(40 - 2a) + 2(a - 10)
N = 5a + 160 - 8a + 2a - 20
N = 140 - a ⑤




Now ,

Looking at ③, since 0 ≤ c ≤ 30,

then 0 ≤ a - 10 ≤ 30

10 ≤ a ≤ 40

But we know 0 ≤ a ≤ 30 .

Combining the two, 10 ≤ a ≤ 30 ⑥



Looking at ④, since 0 ≤ b ≤ 30,

then 0 ≤ 40 - 2a ≤ 30

-40 ≤ -2a ≤ -10

-20 ≤ -a ≤ -5

20 ≥ a ≥ 5 ⑦


Combining ⑥ and ⑦,


10 ≤ a ≤ 20

Since N = 140 - a, and 10 ≤ a ≤ 20,

Then 120 ≤ 140 - a ≤ 130

So 120 ≤ N ≤ 130