# Finding the solution to system of 3 equations with 3 unknowns

• ChiralSuperfields
In summary, the conversation discusses how the equations ##2x_1 - 8x_3 = 4## and ##-2x_1+8x_3= -4## cannot hold at the same time, leading to the conclusion that the solution space is empty and there is nothing more that can be done. The unexpected result of ##0 \ne 1## is accepted and no values for ##x_1## or ##x_3## will satisfy the equation.
ChiralSuperfields
Homework Statement
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Relevant Equations
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For this,

I am trying to find solutions, however, I think I am getting a strange result that I am not too sure how to intercept.

I first multiply the first equation by 2 to get ##2x_1 - 8x_3 = 4## and then I add it to the second equation below to get ##0 = 1##. I think this means that there is no values for ##x_1## or ##x_3## that satisfy the equation. I am not too sure how to go from here but I know ##x_2## is a leading variable of the third equation.

Any help is greatly appreciated

Many thanks!

FactChecker
ChiralSuperfields said:
I first multiply the first equation by 2 to get ##2x_1 - 8x_3 = 4## and then I add it to the second equation below to get ##0 = 1##. I think this means that there is no values for ##x_1## or ##x_3## that satisfy the equation.
Right. If you multiply the first line by ##-2## then you get ##-2x_1+8x_3= -4## and ##-2x_1+8x_3= -3.##
This cannot hold at the same time.
ChiralSuperfields said:
I am not too sure how to go from here but I know ##x_2## is a leading variable of the third equation.

Any help is greatly appreciated

Many thanks!
The solution space is empty. There is nothing left that can be done.

FactChecker and ChiralSuperfields
ChiralSuperfields said:
get ##0 = 1##. I think this means that there is no values for ##x_1## or ##x_3## that satisfy the equation. I am not too sure how to go from here
Good work. Sometimes the hardest problems are the ones that give you an unexpected result. Just double-check your result and accept what it says. Your result is not just about ##x_1## and ##x_3##, it is about ##0 \ne 1##. So nothing more can be done.

ChiralSuperfields and fresh_42

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