Express last equation of system as sum of multiples of first two equations The question says to 'Express the last equation of each system as a sum of multiples of the first two equations." 
System in question being: 
$ x_1+x_2+x_3=1 $
$ 2x_1-x_2+3x_3=3 $
$ x_1-2x_2+2x_3=2 $
The question gives a hint saying "Label the equations, use the gaussian algorithm" and the answer is 'Eqn 3 = Eqn 2 - Eqn 1' but short of eye-balling it, I'm not sure how they deduce that after row-reducing to REF.

You can do row reduction on the transpose:
\begin{align}
\begin{bmatrix}
1 & 2 & 1\\
1 & -1 & -2\\
1 & 3 & 2\\
1 & 3 & 2\\
\end{bmatrix}
&\to
\begin{bmatrix}
1 & 2 & 1\\
0 & -3 & -3\\
0 & 1 & 1\\
0 & 1 & 1\\
\end{bmatrix}
&&\begin{aligned}
R_2&\gets R_2-R_1\\
R_3&\gets R_3-R_1\\
R_4&\gets R_4-R_1
\end{aligned}
\\[6px]
&\to
\begin{bmatrix}
1 & 2 & 1\\
0 & 1 & 1\\
0 & 1 & 1\\
0 & 1 & 1\\
\end{bmatrix}
&&R_2\gets-R_2/3
\\[6px]
&\to
\begin{bmatrix}
1 & 2 & 1\\
0 & 1 & 1\\
0 & 0 & 0\\
0 & 0 & 0\\
\end{bmatrix}
&&\begin{aligned}
R_3&\gets R_3-R_2\\
R_4&\gets R_4-R_2
\end{aligned}
\\[6px]
&\to
\begin{bmatrix}
1 & 0 & -1\\
0 & 1 & 1\\
0 & 0 & 0\\
0 & 0 & 0\\
\end{bmatrix}
&&R_1\gets R_1-2R_2
\end{align}
which makes clear that $C_3=-C_1+C_2$
