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Question 1127259: HI!
How do I solve these linear equations using a matrix?
2/x+5/y+3/z=7
5/x-9/y-7/z=5
3/x+2/y-4/z=-2
Thank you!
Answer by ikleyn(52798)   (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 =  , v =  , w =  . (*)
Then the system takes the form
2u + 5v + 3w = 7 (1)
5u - 9v - 7w = 5 (2)
3u - 2c - 4z = 2 (3)
This system is a standard 3x3-system of linear equations. It is MUCH SIMPLER than the original non-linear system.
This modified system you can solve by the Elimination method or by using the Cramer rule (determinants) - on your choice.
Or, if this standard solution procedure is boring to you, you can use standard solvers from the Internet.
If you decide to go this way, I recommend you the online free of charge solvers at the site www.reshish.com.
In any case, when you complete the solution to the system (1), (2), (3), you must to return to original unknowns
x, y and z using the transformation formulas (*).
To see many problems solved in this way, look into the lesson
- Solving systems of non-linear equations in three unknowns using Cramer's rule
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this online textbook under the topic
"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns"
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.
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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.
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