Question

Solve the following system of equation:
\[\dfrac{5}{{x + 1}} - \dfrac{2}{{y - 1}} = \dfrac{1}{2}\], \[\dfrac{{10}}{{x + 1}} + \dfrac{2}{{y - 1}} = \dfrac{5}{2}\] where \[x \ne - 1\] and \[y \ne 1\]

Answer 1

Hint: In these types of questions substitute \[\dfrac{1}{{x + 1}} = u\] and \[\dfrac{1}{{y - 1}} = v\] then use it to find the values of u and v then use the values of u and v to find the values of x and y.

Complete step-by-step answer:
Let \[\dfrac{1}{{x + 1}} = u\] and \[\dfrac{1}{{y - 1}} = v\]
So the equations are
\[5u - 2v = \dfrac{1}{2}\] and \[10u + 2v = \dfrac{5}{2}\]
Let \[5u - 2v = \dfrac{1}{2}\] equation 1 and \[10u + 2v = \dfrac{5}{2}\] equation 2
Adding equation 1 and equation 2
$ \Rightarrow $\[5u - 2v + 10u + 2v = \dfrac{5}{2} + \dfrac{1}{2}\]
$ \Rightarrow $\[u = \dfrac{1}{5}\]
Putting value of u in equation 1
$ \Rightarrow $\[5 \times \dfrac{1}{5} - 2v = \dfrac{1}{2}\]
$ \Rightarrow $\[v = \dfrac{1}{4}\]
Now using the value of u for finding the value of x
\[\dfrac{1}{{x + 1}} = \dfrac{1}{5}\]
$ \Rightarrow $\[x = 4\]
Now using the value of v for finding the value of y
\[\dfrac{1}{{y - 1}} = \dfrac{1}{4}\]
$ \Rightarrow $\[y = 5\]
Hence the value of x and y are $4,5$.

Note: The equation is in complex form we can simplify the equation by substituting \[\dfrac{1}{{x + 1}} = u\] and \[\dfrac{1}{{y - 1}} = v\] in the equation and then we can compare then we can apply any operation to find the value of u and v since we got the values of variables which we needed to find the value of x and y.