# For which value of k the vector u=[[1],[-2],[k]] is a

For which value of $k$ the vector $u=\left[\begin{array}{c}1\\ -2\\ k\end{array}\right]$ is a linear combination of the vectors ${v}_{1}=\left[\begin{array}{c}3\\ 0\\ -2\end{array}\right]$ and ${v}_{2}=\left[\begin{array}{c}2\\ -1\\ -5\end{array}\right]$.
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Elois Puryear
put the given values in the equation $u=\alpha {v}_{1}+\beta {v}_{2}$
$\left[\begin{array}{c}1\\ -2\\ k\end{array}\right]=\alpha \left[\begin{array}{c}3\\ 0\\ -2\end{array}\right]\beta \left[\begin{array}{c}2\\ -1\\ -5\end{array}\right]$
Now solve the given system get
$3\alpha +2\beta =1$...(i)
$0\alpha -\beta =-2$...(ii)
$-2\alpha -5\beta =k$...(iii)
Now find the value of $\alpha ,\beta$ using equations i and ii
From equation (ii)
$-\beta =-2$
$\beta =2$
Put this value in equation i and solve for $\alpha$
$3\alpha +2\left(2\right)=1$
$3\alpha +4=1$
$3\alpha =-3$
$\alpha =-1$
Now put the value of $\alpha ,\beta$ in equation (iii) and solve it for k
$-2\left(-1\right)-5\left(2\right)=k$
$2-10=k$
$k=-8$
Value of k is -8