Question

solve the following pair of equation by reducing them to a pair of linear equations 5/x+y +1/x-y =2 and 10/x+y+3/x-y​

Answer 1

The Solution is {x,y} = {3,2}

Step-by-step explanation:

Question is Incomplete Complete question is given below.

Solve the following pair of linear equation by reducing them to a pair of linear equation 5/x+y +1/x-y = 2 10/x+y + 3/x-y =5.

Given:

$\frac{5}{x+y}+\frac{1}{x-y}=2\\\\\frac{10}{x+y}+\frac{3}{x-y}=5$

We need to find the solutions for the equation.

Solution:

Let us consider $\frac{1}{x+y} = a$ and $\frac{1}{x-y}=b$

Now the equation becomes as

$5a+b=2$ ⇒ equation 1

$10a+3b=5$ ⇒ equation 2

Now We solve the above equations.

First we will multiply equation 1 by 3 we get;

$3(5a+b)=2\times3$

$15a+3b = 6$  ⇒ equation 3

Now we will subtract equation 2 from equation 3 we get;

$15a+3b-(10a+3b)=6-5\\\\15a+3b-10a-3b=1\\\\5a=1\\\\a= \frac{1}{5}$

Substituting the value of a in equation 1 we get;

$5a+b=2\\\\5\times \frac{1}{5}+b=2\\\\1+b=2\\\\b=2-1\\\\b=1$

Now we will substitute the value of a and b we get;

$\frac{1}{x+y}=\frac{1}{5}$

By cross multiplying we get;

$x+y=5$ ⇒ equation 4

$\frac{1}{x-y} =1$

By cross multiplying we get;

$x-y=1$ ⇒ equation 5

Adding equation 4 and equation 5 we get;

$x+y+x-y=5+1\\\\2x=6\\\\x=\frac{6}{2}\\\\x=3$

Substituting the value of x in equation 4 we get;

$x+y=5\\\\3+y=5\\\\y =5-3\\\\y =2$

Hence the Solution is {x,y} = {3,2}