confirmation of method: Write the vector $(2,-3)$ as a linear combination of vectors $(-1,4)$ and $(1,2)$ Write the vector $(2,-3)$ as a linear combination of vectors $(-1,4)$ and $(1,2)$
this was on an assignment rather hw so I want to be sure I'm using the right method and have the right answer. 
I confirm $w = k_1u + k_2v$ to prove I have the right linear combination of vectors.
I solve for $k_1$ and $k_2$
$(-1,4)k_1 + (1,2)k_2 = w$
guassian elimination
$|-1,4 = 2|$
$| 1,2 = -3|$
$=$
$|-1,4 = 2|$
$| 0,6 = -1|$
back substitution
$k_2 = -1/6$
$-1k_1 = 6/3+2/3$
$k_1 = -8/3$
plug $k$'s into vector equation
$(-1,4)(-1/6) + (1,2)(-8/3) = w$
$(1/6,2/3) + (-8/3,-16/3) = w$
$-15/6,-14/3 = w$ .....nope... where did I do it wrong.. and am I using the right method?

It shouldn't be this matrix
$$
\left(
\begin{array}{cc|c}
-1 & 4 & 2 \\
1 & 2 & -3
\end{array}
\right)
$$
since this corresponds to the system of equations
\begin{align*}
-k_1+4k_2 &= 2 \\
k_1+2k_2 &= -3 \\
\end{align*}
whereas you want the system of equations
\begin{align*}
-k_1+k_2 &= 2 \\
4k_1+2k_2 &= -3 \\
\end{align*}
to solve $k_1(-1,4)+k_2(1,2)=(2,-3)$.
So it should instead be
$$
\left(
\begin{array}{cc|c}
-1 & 1 & 2 \\
4 & 2 & -3
\end{array}
\right)
$$
Aside from that, the method should work fine.
