<s>[INST] Find the value of $a$ for which v is in the set H = span 
Find the value of $a$ for which $ v = \begin{bmatrix}10\\-6\\6\\a\end{bmatrix}$ is in the set $H = span \{ \begin{bmatrix}-5\\2\\-5\\-3\end{bmatrix}, \begin{bmatrix}0\\-2\\-1\\1\end{bmatrix}, \begin{bmatrix}0\\0\\-3\\-1\end{bmatrix}\}$

$a = ?$
I found: 
$10 = -5x \implies x = -2$
$-6 = 2x - 2y \implies -4 - 2(1) \implies y = 1$
$6 = -5x - y -3z \implies -5(-2) - 1 - 3z \implies z = 1$
$a = 3x + y - z \implies a = -6+1-1 = -6$ but I keep getting the answer wrong, how is $a \ne -6$?
 [/INST] Note that the augmented system
$$
\left[
\begin{array}{rrr|r}
-5 & 0 & 0 & 10 \\
2 & -2 & 0 & -6 \\
-5 & -1 & -3 & 6 \\
-3 & 1 & -1 & a
\end{array}
\right]
$$
can be row-reduced with the following steps:


*

*scale row 1 by $-1/5$

*add $-2$ times row 1 to row 2

*add $5$ times row 1 to row 3

*add $3$ times row 1 to row 4

*scale row 2 by $-1/2$

*add $1$ times row 2 to row 3

*add $-1$ times row 2 to row 4

*scale row 3 by $-1/3$

*add $1$ times row 3 to row 4


Performing these operations gives
$$
\DeclareMathOperator{rref}{rref}\rref
\left[\begin{array}{rrr|r}
-5 & 0 & 0 & 10 \\
2 & -2 & 0 & -6 \\
-5 & -1 & -3 & 6 \\
-3 & 1 & -1 & a
\end{array}\right]
=
\left[\begin{array}{rrr|r}
1 & 0 & 0 & -2 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & a - 6
\end{array}\right]
$$
This shows that $v\in H$ if and only if $a=6$.
</s>