I'm reading the Linear Algebra book by Jim Hefferon and I've encountered a difficult question to me. I don't understand the intention of the problem, therefore I don't know where to start solving it.
Can we derive $6x - 9y + 5z = -2$ by a sequence of Gaussian reduction steps from the equations in the system?
$2x + y - z = 4$
$6x - 3y + z = 5$
The intention is quite straightforward: it is important to understand if an equation is independent from a system of equations or a logical consequence of it.
The problem is equivalent to the following question: is the vector $w=(6,-9,5,-2)$ a linear combination of the vectors $u=(2,1,-1,4)$ and $v=(6,-3,1,5)$. In other words, are there real numbers $x,y$ such that $w=xu+yv$. This leads to the following system of linear equations:
$2x+6y=6$
$x-3y=-9$
$-x+y=5$
$4x+5y=-2$
This has a unique solution: $x=-3, y=2$. So yes, the equation can be expressed from the system: multiply the first equation by $-3$ and the second by $2$, and add them up.
(If the above system of linear equations had no solution, it would mean that your equation is not expressible from the system. Think about it.)