if a b and c are positive numbers satisfying a+1/b=4,b+1/c=1 and c+1/a=7/3 then the value of 6abc.
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Step-by-step explanation:
We need to form three simultaneous equations:
a + 1/b = 4
a + 1 = 4b
4b - a = 1...............i)
b + 1/c = 1
b + 1 = c
c - b = 1 ....................ii)
c + 1 / a = 7/3
3(c + 1) = 7a
3c + 3 = 7a
3c - 7a = 3..................iii)
From these equations we can solve these equations simultaneously to get a , b and c.
From 1 we can get by substitution.
a = 4b - 1...**
Substitute this in 3 .
3c - 7(4b - 1) = -3
3c - 28b + 7 = -3
3c = -3 - 7 + 28b
3c = -10 + 28b
c = -10/3 + 28b/3
We substitute this in 2 to get :
-10/3 + 28b/3 - b = 1
Multiply through by 3:
-10 + 28b - 3b = 3
25b = 13
b = 13/25
We can get a from ***
a = 4 × 13/25 - 1 = 27/25
From 2: c = 1 + b
c = 38/25
The value of 6abc = 6 × 38/25 × 13/25 × 27/25 = 5.122
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