Question 1093252
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There is a standard method to solve such non-linear systems.


<pre>
Introduce new variables  u = 1/x  and  v = 1/y.

Then the original system of non-linear equations becomes (by a magical way) a linear system relative new variables "u" and "v".

3u + 4v = 5/2,   (1)
6u +  v = 1/2.   (2)


Now you can apply any method you want / you know to solve (1), (2).

I will use Elimination method. 

Multiply equation (1) by 4 (both sides), and equation (2) by 2 (both sides).


You will get

12u + 16v = 10,   (3)
12u +  2v = 1.    (4)


Subtract eq(4) from eq(3) to get

14v = 9.

Then  v = 9/14.

Having this, you easily get (from (4))  12u = 1 - 2*(9/14) = 1 - 18/14 = -4/14 = -2/7.

Hence,  u = -2/(7*12) = -1/42.

Thus you get  u = -1/42,  v = 9/14.


Now recall how you introduced u and v:  u = 1/x = -1/42;  hence,  x = -42.

                                   and   v = 1/y = 9/14;  hence,  y = 14/9.

The original system solved.  The solutions are  x = -42  and  y = 14/9.
</pre>


You can substitute it into the original equations to check it on your own (I did it).



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To see more tricks of this kind, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A &nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-two-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in two unknowns using the Cramer's rule</A> 

in this site.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
&nbsp;&nbsp;&nbsp;&nbsp; "<U>2x2-Matrices, determinants, Cramer's rule for systems in two unknowns</U>" 



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.