Question

If \[x=by+cz+du\], \[y=cz+ax+du\], \[z=ax+by+du\], \[u=ax+by+cz\]. Then find the value of the expression \[\dfrac{a}{1+a}+\dfrac{b}{1+b}+\dfrac{c}{1+c}+\dfrac{d}{1+d}\] is
(a)2
(b)\[\dfrac{1}{2}\]
(c)1
(d)3

Answer 1

Hint: Add all the 4 expressions together. Compare the LHS and RHS and make them equal by finding the values of a, b, c and d. Substitute the same in the given expression and find its value.

Complete step-by-step answer:
We have been given four expressions, which are,
\[\Rightarrow x=by+cz+du\] - (1)
\[\Rightarrow \] \[y=cz+ax+du\] - (2)
\[\Rightarrow \] \[z=ax+by+du\] - (3)
\[\Rightarrow \] \[u=ax+by+cz\] - (4)
Now let us add all these (4) expressions together,
\[\Rightarrow x+y+z+u=3ax+3by+3cz+3du\] - (5)
Let us compare the LHS and RHS of expression given above.
The co – efficient of the expression are of the form. If we put a = b = c = d = \[\dfrac{1}{3}\] in (5) we get,
\[\Rightarrow x+y+z+u=3ax+3by+3cz+3du\]
\[\begin{align}
  & \Rightarrow x+y+z+u=3\times \dfrac{1}{3}x+3\times \dfrac{1}{3}y+3\times \dfrac{1}{3}z+3\times \dfrac{1}{3}u \\
 & \Rightarrow x+y+z+u=x+y+z+u \\
\end{align}\]
From this we can see that, LHS = RHS.
We have been asked to find the value of the expression,
\[\Rightarrow \dfrac{a}{1+a}+\dfrac{b}{1+b}+\dfrac{c}{1+c}+\dfrac{d}{1+d}\]
Now put a = b = c = d = \[\dfrac{1}{3}\] in the above expression.
As a = b = c = d, we can change the expression as,
\[\begin{align}
  & \Rightarrow \dfrac{a}{1+a}+\dfrac{b}{1+b}+\dfrac{c}{1+c}+\dfrac{d}{1+d}=\dfrac{a}{1+a}+\dfrac{a}{1+a}+\dfrac{a}{1+a}+\dfrac{a}{1+a} \\
 & \Rightarrow \dfrac{a}{1+a}+\dfrac{b}{1+b}+\dfrac{c}{1+c}+\dfrac{d}{1+d}=4\times \left( \dfrac{a}{1+a} \right) \\
\end{align}\]
Now put a = \[\dfrac{1}{3}\]
\[\Rightarrow 4\left[ \dfrac{\dfrac{1}{3}}{1+\dfrac{1}{3}} \right]=4\left[ \dfrac{\dfrac{1}{3}}{\dfrac{3+1}{3}} \right]=4\left[ \dfrac{\dfrac{1}{3}}{\dfrac{4}{3}} \right]=4\left[ \dfrac{1}{3}\times \dfrac{3}{4} \right]=4\times \dfrac{1}{4}=1\]
Hence we got the value of the expression as 1.
i.e. \[\dfrac{a}{1+a}+\dfrac{b}{1+b}+\dfrac{c}{1+c}+\dfrac{d}{1+d}=1\]
\[\therefore \] Option (c) is the correct answer.

Note: Don’t try to substitute the 4 expressions in one another seeing the similar terms in the expressions. It is the first thing that comes to your mind, but it makes the solution complex. Thus add them together and compare RHS and LHS.