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For a certain value of k, the system

x + y + 3z = 10

-4x + 8y + 5z = 7

kx + z = 3

has no solutions.  What is this value of k?

 Feb 7, 2023
 #1
avatar+2666 
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Take the first equation, and multiply it by 8. This gives us \(8x + 8y + 24x = 80\)

 

Subtracting the second equation from this gives us \(12x + 19z = 73\)

 

Now, take the third equation and multiply it by 19. This gives us \(19kx + 19z = 57\)

 

These two equations won't have any solutions when the coefficient of x in both equations is equal. This will mean that two of the same things sum to two different values, which is impossible. 

 

This means that \(12 = 19k\), meaning \(k = \color{brown}\boxed{12 \over 19}\)

 Feb 7, 2023

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