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For a certain value of $k$, the system \begin{align*} x + y + 3z &= 10, \\ -4x + 2y + 5z &= 7, \\ kx + z &= 3 \end{align*} has no solutions. What is this value of $k$?

Guest Nov 12, 2017
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x + y + 3z  = 10

-4x + 2y + 5z = 7

kx + z  = 3   ⇒  z = 3 - kx  (3)

Sub (3) into the first two equations

x + y + 3 [ 3 - kx]  = 10

-4x + 2y + 5[3 - kx ] = 7    simplify   these

(1 - 3k) x   + y  =  1

(-4 -5k)x   + 2y  = - 8

This will have no soution for k when this is true

( 1 - 3k) * 2  -  ( - 4 - 5k)* 1  = 0

2 - 6k + 4 + 5k  =  0

6  =  k

CPhill  Nov 12, 2017

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