SOLUTION: Hi! How do I solve these linear equations using a matrices? 6/x+1/y=1 9/x-2/y=5 Thank you!

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Question 1127257: Hi!
How do I solve these linear equations using a matrices?
6/x+1/y=1
9/x-2/y=5
Thank you!
Answer by ikleyn(52806) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I'd like to start noticing that the given system IS NOT a linear: it is the system of NON-LINEAR equations. 


The standard method to solve such a system is to introduce new variables

u = 1%2Fx,  v = 1%2Fy.


Then the system takes the form


6u + 1v = 1     (1)
9u - 2v = 5     (2)


This you can solve by the Substitution method, for example.


For it express v = 1 - 6u  from (1) and substitute into eq(2).  You will get


9u - 2*(1-6u) = 5,

9u - 2 + 12u = 5,

21u = 5 + 2 = 7,

u = 7%2F21 = 1%2F3.


Then from eq(1),  v = 1 - 6u = 1+-+6%2A%281%2F3%29 = 1 - 2 = -1.


Now,  x = = 1%2Fu = 1%2F%28%281%2F3%29%29 = 3  and  y = 1%2Fv = 1%2F%28-1%29 = -1.     


ANSWER.  x = 3  and  y = -1.

Solved.

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Again, the lesson to learn from the solution is THIS :

            The standard method to solve such a system is to introduce new variables.

            In this way, you reduce the original system of non-linear equations to the standard system of linear equations,
            for which you can use many different and well known standard methods.