Solve System of 4 Equations: Find x,y,u,v in Terms of a,b,c,d

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The discussion centers on solving a system of four equations to express variables x, y, u, and v in terms of a, b, c, and d. It concludes that the determinant of the right-hand side is zero, indicating that the equations are not independent. The relationships established are u/x = 2a, u/y = 2b, v/x = 2c, and v/y = 2d, demonstrating that knowing a, b, c, and d only provides the ratios of u and v to x and y. Therefore, a, b, c, and d do not uniquely determine x, y, u, and v.

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Bruno Tolentino
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Given this system of equations:

?temp_hash=e1398efa2daaad05acbf8f3c26412522.png


I want to write x,y,u,v in terms of a,b,c,d. Is possible?
 

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Take the determinant on the righthand side: it is zero. That means the equations are not independent.
 
you are asking if a,b,c,d determine x,y,u,v. But clearly u/x = 2a, u/y = 2b, v/x = 2c, and v/y = 2d, so if you know a,b,c,d, you only know the ratios of u and v to x and y. so you could double u and v and x and y, and the same ratios would hold. So a,b,c,d do not determine x,y,u,v.
 
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