Help with solving set of equations

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To solve the set of six linear equations with six variables (Aup, Adown, Bup, Bdown, Cup, and Cdown), Gaussian elimination is recommended as an effective method. This technique systematically reduces the equations to a simpler form, making it easier to isolate and solve for each variable. Users can apply matrix representation of the equations to facilitate the elimination process. Additionally, utilizing computational tools or software can streamline the calculations involved. Overall, Gaussian elimination is a reliable approach for solving such systems of equations.
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I have 6 equations and 6 variables I want to solve for:

[PLAIN]http://people.virginia.edu/~km2mr/Untitled.jpg

What would be the best way to go about solving for Aup, Adown, Bup, Bdown, Cup, and Cdown.

Thanks.
 
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The equations are linear in these 6 variables, so Gaussian elimination should do it.
 
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