Question

Solve the following system of equation:
\[\dfrac{{22}}{{x + y}} + \dfrac{{15}}{{x - y}} = 5\], \[\dfrac{{55}}{{x + y}} + \dfrac{{45}}{{x - y}} = 14\]

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.

Complete step-by-step answer:
Let \[\dfrac{1}{{x + y}} = u\] and \[\dfrac{1}{{x - y}} = v\]
So the equations are
\[22u + 15v = 5\] and \[55u + 45v = 14\]
Let \[22u + 15v = 5\] equation 1 and \[55u + 45v = 14\] equation 2
Multiplying equation 1 by 3 and equation 2 by 1
$ \Rightarrow $\[3 \times 22u + 3 \times 15v = 3 \times 5\] and \[1 \times 55u + 1 \times 45v = 1 \times 14\]
$ \Rightarrow $\[66u + 45v = 15\] and \[55u + 45v = 14\] subtracting both the equations with each other
$ \Rightarrow $\[66u + 45v - 55u - 45v = 15 - 14\]
$ \Rightarrow $\[u = \dfrac{1}{{11}}\]
Putting value of u in equation 1
$ \Rightarrow $\[22 \times \dfrac{1}{{11}} + 15v = 5\]
$ \Rightarrow $\[v = \dfrac{1}{5}\]
Now using the value of u for finding the value of x
\[\dfrac{1}{{x + y}} = \dfrac{1}{{11}}\]
$ \Rightarrow $\[x + y = 11\] (Equation 3)
Now using the value of v for finding the value of y
\[\dfrac{1}{{x - y}} = \dfrac{1}{5}\]
$ \Rightarrow $\[x - y = 5\] (Equation 4)
Adding equation 3 and 4
$ \Rightarrow $\[x - y + x + y = 11 + 5\]
$ \Rightarrow $\[x = 8\]
Putting the value of x in equation 4
$ \Rightarrow $\[8 - y = 5\]
$ \Rightarrow $\[y = 3\]
Hence the value of x and y are 8, 5.

Note: In the given question first substitute \[\dfrac{1}{{x + y}} = u\]and \[\dfrac{1}{{x - y}} = v\] then find the values of u and v make the equations 1 and equation 2 and then multiply the equation 1 by 3 and equation 2 by 1 and subtract both the equations with each other and find the value of u and v after obtaining the values of u and v then put the values of u and v in equations \[\dfrac{1}{{x + y}} = u\] and \[\dfrac{1}{{x - y}} = v\] make the equations 3 and 4 then with the help of equations find the value of x and y.