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There exists a real number k such that the equation \(\begin{pmatrix} -3 \\ \phantom -0 \end{pmatrix} + t \begin{pmatrix} -4 \\ \phantom -1 \end{pmatrix} = \begin{pmatrix} -4 \\ \phantom -k \end{pmatrix} + s \begin{pmatrix} \phantom -8 \\ -2 \end{pmatrix}\)
has infinitely many solutions in t and s. Find k.

 Feb 9, 2020
 #1
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-1

The value of k works out to -7.

 Feb 10, 2020
 #2
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+4

The two equations are:

 

-3 - 4t = -4 + 8s    (1)       and   0 + t = k - 2s    (2)

 

Rearrange these as follows:

 

from (1):   t = 1/4 - 2s

from (2):   t = k - 2s

 

So k = 1/4

 Feb 10, 2020

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