Can the equation $5x-3y=2$ be derived, by a sequence of Gaussian reduction steps, from the equations in this system?
\begin{array}{*{2}{rc}r}2x&+&2y&=&5\\ 3x&+&y&=&4\end{array}
This is taken from the Linear Algebra textbook on Wikibooks (problem 10, 1.1 Linear Systems)
The worked solution given is "No. The given equation is satisfied by the pair (1,1). However, that pair does not satisfy the first equation in the system."
Can someone please explain why? The way the solution is given, it's as if any ($x,y$) that satisfies $5x-3y=2$ must also satisfy any of the two equations in the given system of equations for there to be a sequence of Gaussian reduction steps to reach $5x-3y=2$. But I don't see that.
What I do know is that the theorem in the same page explains that any system of equation derived from another through Gaussian operations must have the same set of solutions. But in this case, there is only 1 equation, not a system of equations.
Gaussian elimination corresponds to linear combination of the equations in order to simplify a given linear system, in you example
$$\begin{array}{*{2}{rc}r}2x&+&2y&=&5\\3x&+&y&=&4\end{array}$$
we can multiply the first equation by 3 and the second by 2 and then subtract the first equation from the second to obtain
$$\begin{array}{*{2}{rc}r}6x&+&6y&=&15\\6x&+&2y&=&8\end{array}$$
and then
$$\begin{array}{*{2}{rc}r}2x&+&2y&=&5\\&&-4y&=&-7\end{array}$$
and the latter system has the same solutions of the original one.