Question

Solve the following pair of linear equations : \[\dfrac{{10}}{{x + y}} + \dfrac{2}{{x - y}} = 4\] and \[\dfrac{{15}}{{x + y}} - \dfrac{5}{{x - y}} = - 2\] .

Answer 1

Hint: We have to find the values of the variables \[x\] and \[y\] by solving the given pair of linear equations . We solve this question using the concept of solving the pair of linear equations by elimination method or any other method for solving the linear equations like the substitution method .First , we will substitute the value of the relation between \[x\] and \[y\] with two other variables . From this we will obtain two linear equations which we can solve using the elimination method . Then putting back the values in the assumed relation and again applying elimination method , we can compute the values of the variables \[x\] and \[y\] .

Complete step-by-step answer:
Given :
\[\dfrac{{10}}{{x + y}} + \dfrac{2}{{x - y}} = 4 - - - - (1)\]
\[\dfrac{{15}}{{x + y}} - \dfrac{5}{{x - y}} = - 2 - - - - (2)\]

Now , let us consider that the relation between \[x\] and \[y\] be such that as given below :
\[\dfrac{1}{{x + y}} = u\] and \[\dfrac{1}{{x - y}} = v\]

Now , we will substitute these values in equation \[(1)\] and \[(2)\]
After substituting the values , we get the equations as
\[10u + 2v = 4 - - - - - (3)\]
\[15u - 5v = - 2 - - - - - (4)\]

Now , we will solve the equation \[(3)\] and \[(4)\] using elimination method for the value of \[u\] .
For solving the equations :
Multiplying equation \[(3)\] by \[\dfrac{5}{2}\] , we get
\[\dfrac{5}{2}\left[ {10u + 2v = 4} \right]\]
\[25u + 5v = 10 - - - (5)\]

Now , on adding equation \[(5)\] and \[(4)\] we get the value of u as :
\[25u + 5v + 15u - 5v = 10 - 2\]
\[40u = 8\]

On further simplifying we get the value of \[u\] as :
\[u = \dfrac{1}{5}\]

Now , putting value of \[u\] in equation \[(4)\] , we get
\[15 \times \dfrac{1}{5} - 5v = - 2\]
\[3 - 5v = - 2\]
Further , we get
\[ - 5v = - 2 - 3\]

On further simplifying we get the value of \[v\] as :
\[v = 1\]

Now , substituting the values of \[v\] and \[u\]in the relation between \[x\] and \[y\] , we get
\[\dfrac{1}{{x + y}} = \dfrac{1}{5} - - - (6)\]
\[\dfrac{1}{{x - y}} = 1 - - - (7)\]

Taking the reciprocals of the above we get the two equations as
\[x + y = 5 - - - (6)\]
\[x - y = 1 - - - (7)\]

Now we will be solving equation \[(6)\] and \[(7)\]
Adding the equation \[(7)\] and \[(6)\] , we get
\[x + y + x - y = 5 + 1\]
\[2x = 6\]

On solving the equations , we get the value of \[x\] as
\[x = 3\]

Putting the value of \[x\] in equation \[(6)\] , we get
\[3 + y = 5\]
\[y = 2\]

Hence , on solving the pair of linear equation \[\dfrac{{10}}{{x + y}} + \dfrac{2}{{x - y}} = 4\] and \[\dfrac{{15}}{{x + y}} - \dfrac{5}{{x - y}} = - 2\] we get \[x = 3\] and \[y = 2\] .

Note: We can solve any of the pairs of linear equations using the various methods . We use the method of elimination as it is a bit easier than the other methods for solving the equations . We will get the same value for the solutions of the pair of linear equations using any of the methods i.e. elimination method , cross – multiplication method , substitution method or any other .