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avatar+174 

For a certain value of $k$, the system
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y

has no solutions. What is this value of $k$?

 Jun 12, 2024
 #1
avatar+129733 
+1

Simplify as

 

x + y + 3z = 10

7x + 9y + z  = 18

kx + 14y +7z = 8

 

If the system has no solutions, the determinant =  0

 

So

 

1  1   3   1   1

7  9   1   7   9

k  14  7  k   14

 

[ 1*9*7 + 1*1 *k + 3*7 *14 ] - [ 9*3*k + 14 * 1 * 1 + 7*7*1 ]  = 0

 

[ 63 + k + 294] - [ 27k + 14 + 49 ] =  0

 

294 - 26k   = 0

 

k = 294/26   =147 / 13

 

 

cool cool cool

 Jun 13, 2024

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