SOLUTION: x=by+cz y=cz+ax z=ax+by prove a/1+a + b/1+b + c/1+c = 1 pls help with solution...thanks

Question 472703: x=by+cz
y=cz+ax
z=ax+by
prove a/1+a + b/1+b + c/1+c = 1
pls help with solution...thanks
Found 2 solutions by richard1234, robertb:
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Adding all three of our given statements we obtain x+y+z = 2ax + 2by + 2cz. Matching the x,y,z coefficients, we get 2a = 1, 2b = 1, 2c = 1, so a,b,c = 1/2. Replacing 1/2 into a/(1+a) + b/(1+b) + c/(1+c) yields a sum of 1.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Subtract the 2nd equation from the 3rd: z - y = by - cz, whence z(1+c) = y(1+b).
Similarly, x(1+a) = y(1+b).
==> x(1+a) = y(1+b) = z(1+c).
==> and
Subsitute into the 1st equation: .
Since x is not identically 0, we can cancel x throughout.
==> .
==>
==>
The conclusion follows after one more step.

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