if - a(y+z)=x b(z+x)=y c(x+y)=z then prove that a^1+a+b^1+b+c^1+c = 1
Answers
Answer:
From question we get,
⇒ a=
y+z
x
⇒ b=
z+x
y
⇒ c=
x+y
z
Now,
ab=
(y+z)(z+x)
xy
bc=
(z+x)(x+y)
yz
ca=
(y+z)(x+y)
xz
abc=
(y+z)(x+y)(x+z)
xyz
L.H.S.=bc+ca+ab+2abc
=ab+bc+ca+2abc
=
(y+z)(z+x)
xy
+
(z+x)(x+y)
yz
+
(y+z)(x+y)
xz
+
(y+z)(x+y)(x+z)
xyz
=
(y+z)(x+y)(x+z)
xy(x+y)+yz(y+z)+xz(x+z)+2xyz
=
(y+z)(x+y)(x+z)
x
2
y+xy
2
+y
2
z+z
2
y+x
2
z+z
2
x+2xyz
=
(y+z)(x+y)(x+z)
y(x
2
+2xz+z
2
)+y
2
(x+z)+zx(x+z)
=
(y+z)(x+y)(x+z)
y(x+z)
2
+y
2
(x+z)+zx(x+z)
=
(y+z)(x+y)(x+z)
(x+z)[y(x+z)+y
2
+zx]
=
(y+z)(x+y)(x+z)
(x+z)(yx+yz+y
2
+zx)
=
(x+z)(x+y)(x+z)
(x+z)[y(x+y)+z(x+y)]
=
(y+z)(x+y)(x+z)
(x+z)(x+y)(y+z)
=1
=R.H.S.