Let $a_1=\begin{bmatrix}1\\5\\-1\end{bmatrix},\ a_2=\begin{bmatrix}-6\\-26\\2\end{bmatrix}$, and $b=\begin{bmatrix}5\\9\\h\end{bmatrix}$. For what value(s) of h

Let $a_1=\begin{bmatrix}1\\5\\-1\end{bmatrix},\ a_2=\begin{bmatrix}-6\\-26\\2\end{bmatrix}$, and $b=\begin{bmatrix}5\\9\\h\end{bmatrix}$. For what value(s) of h is b in the plane spanned by $$\displaystyle{a}_{{1}}$$ and $$\displaystyle{a}_{{2}}$$?

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Ralph Lester
Given:
$a_1=\begin{bmatrix}1\\5\\-1\end{bmatrix},\ a_2=\begin{bmatrix}-6\\-26\\2\end{bmatrix},\ and\ b=\begin{bmatrix}5\\9\\h\end{bmatrix}$
To find value of h if b spanned by $$\displaystyle{a}_{{1}}$$ and $$\displaystyle{a}_{{2}}$$
Since b is spanned by $$\displaystyle{a}_{{1}}$$ and $$\displaystyle{a}_{{2}}$$
$$\displaystyle\alpha{\left({1},{5},-{1}\right)}+\beta{\left(-{6},-{26},{2}\right)}={\left({5},{9},{h}\right)}$$
$$\displaystyle\alpha-{6}\beta={9}$$
$$\displaystyle{5}\alpha-{26}\beta={9}$$
$$\displaystyle-\alpha+{2}\beta={h}$$
Solving first two equation, we get
$$\displaystyle\alpha=-{19},\ \beta=-{4}$$
$$\displaystyle{h}={11}$$