Grade 7Algebra
If abcd = 1, and abc + ab + a + 1 ≠ 0, show a/(abc+ab+a+1) + b/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1) = 1
If abcd = 1, and abc + ab + a + 1 ≠ 0, show a/(abc+ab+a+1) + b/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1) = 1
2 Answers
The Titans FLL Team
what you do is that you set each of a, b, c and d =1. this can show a/(abc+ab+a+1) + b/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1) = 1
Aditya Gupta
a/(abc+ab+a+1) + b/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1)
= a/(abc+ab+a+abcd) + b/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1)
= 1/(bcd+bc+b+1) + b/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1)
= (1+b)/(bcd+bc+b+1) + c/(cda+cd+c+1) + d/(dab+da+d+1)
= (1+b)/(bcd+bc+b+1) + bc/(cdab+cdb+cb+b) + bcd/(dabcb+dacb+dcb+cb)
= (1+b)/(bcd+bc+b+1) + bc/(bcd+bc+b+1) + bcd/(dcb+cb+b+1)
= (1+b+bc+bcd)/(bcd+bc+b+1)
= 1
actually, you can also solve this by simply substituting a= 1/bcd and then simplifying. both methods are equally good.
kindly approve :))
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