Question

Solve the following system of equation:
\[\dfrac{5}{{x + y}} - \dfrac{2}{{x - y}} = - 1\], \[\dfrac{{15}}{{x + y}} + \dfrac{7}{{x - y}} = 10\]

Answer 1

Hint: In these types of questions substitute \[\dfrac{1}{{x + y}} = u\] and \[\dfrac{1}{{x - y}} = v\] 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.
Let \[\dfrac{1}{{x + y}} = u\] and \[\dfrac{1}{{x - y}} = v\]

Complete step-by-step answer:
So the equations are
\[5u - 2v = - 1\] and \[15u + 7v = 10\]
Let \[5u - 2v = - 1\] equation 1 and \[15u + 7v = 10\] equation 2
Multiplying equation 1 by 3 and equation 2 by 1
$ \Rightarrow $\[3 \times 5u - 3 \times 2v = 3 \times ( - 1)\] and \[1 \times 15u + 1 \times 7v = 1 \times 10\]
$ \Rightarrow $\[15u - 6v = - 3\] and \[15u + 7v = 10\] subtracting both the equations with each other
$ \Rightarrow $\[15u + 7v - 15u + 6v = 10 - ( - 3)\]
$ \Rightarrow $\[v = 1\]
Putting value of v in equation 1
$ \Rightarrow $\[5u - 2 \times 1 = - 1\]
$ \Rightarrow $\[u = \dfrac{1}{5}\]
Now using the value of u for finding the value of x
\[\dfrac{1}{{x + y}} = \dfrac{1}{5}\]
$ \Rightarrow $\[x + y = 5\] (Equation 3)
Now using the value of v for finding the value of y
\[\dfrac{1}{{x - y}} = 1\]
$ \Rightarrow $\[x - y = 1\] (Equation 4)
Adding equation 3 and 4
$ \Rightarrow $\[x + y + x - y = 5 + 1\]
$ \Rightarrow $\[x = 3\]
Putting the value of x in equation 4
$ \Rightarrow $\[3 - y = 1\]
$ \Rightarrow $\[y = 2\]
Hence the value of x and y are 3 and 2 respectively.

Note: The equation is in complex form we can simplify the equation by substituting \[\dfrac{1}{{x + y}} = u\] and \[\dfrac{1}{{x - y}} = v\] 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.