Question 1036350
Solve for x and y
15/x-y + 22/x+y = 5
40/x-y + 55/x+y =13
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<pre>
{{{15/(x-y) + 22/(x+y)}}} = {{{5}}},    (1)
{{{40/(x-y) + 55/(x+y)}}} = {{{13}}}.   (2)

The standard way to solve problems like this, is to introduce new variables

u = {{{1/(x - y)}}},    (3)
v = {{{1/(x + y)}}}.    (4)

Then the system of non-linear equations (1) and (2) becomes the system of linear equations

15u + 22v =  5,    (5)
40u + 55v = 13.    (6)

To solve it, apply the Elimination method. Multiply equation (5) by 8 (both sides) and multiply the equation (6) by 3, 
then distract. You will get

176v - 165v = 40 - 39   --->  11v = 1   --->  v = {{{1/11}}}.

Now, from (5)  

15u = 5 - 22v = {{{5 - 22/11}}}  --->   15u = 3   --->   u = {{{3/15}}} = {{{1/5}}}.

Returning back to the original unknowns x and y, you have

x - y =  5,    (7)
x + y = 11.    (8)

Now we are at the finish line. Add (7) and (8), and you will get

2x = 16   --->  x = {{{16/2}}} = 8. 

From (8),  y = 11 - x = 11 - 8 = 3.

<U>Answer</U>. x = 8, y = 3.
</pre>

The lesson to learn is the method of introducing these new variables.