If a(b+c) = 32, b(c+a) = 65, c(a+b) = 77 then find the value of a*b*c

Question

If a , b , c are positive real numbers such that

a(b+c) = 32
b(c+a) = 65
c(a+b) = 77

Find the value of a*b*c

Answer 1

a(b+c) = 32 .....ab + ac = 32 so I get bc - ac = 33
b(c+a) = 65.......bc + ab = 65 ac - ab = 12
c(a+b) = 77.......ac + bc = 77 bc - ab = 45

so, ac + bc + bc - ac = 77 + 33 so, bc = 55
ab + ac + ac - ab = 32 + 12 ac = 22
bc + ab - bc + ab = 65 - 45 ab = 10

so a*b*c = squareroot of ( 10 * 22 * 55 ) = 110 .

Answer 2

Step 1: Solving the equation gives
ab = 10
bc = 55
ac = 22

Step 2: multiply these three equations
ab*bc*ac = 10*55*22

Step 3: taking the squareroot
abc = 110