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avatar+190 

For a certain value of $k$, the system
\begin{align*}
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y
\end{align*}
has no solutions.  What is this value of $k$?

 Dec 16, 2023
 #1
avatar+126978 
+1

Rearrange as

x + y + 3z = 10

7x + 9y + z = 18

kx + 14y + 7z  =  8

 

Multiply the second equation through by -3  = -21x -27y -3z = -54

Add to  the first

-20x - 26y =  -44        (1)

 

Multiply the second equation by -7  = -49x - 63y -7z = -126

Add to the third

(-49 + k)x -49y = -118   (2)

 

Multiply (1) by -49    and (2)  by -26

 

980x + 1274y = 2156

-26(-49 + k)x +  1274y = 3068 

 

To have no solution

980 = -26(-49 + k)

980 = 1274 - 26k

26k = 1274 - 980

26k = 294

k = 294 / 26 =   147 / 13

 

cool cool cool

 Dec 16, 2023

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