Can anybody explain to me what I did wrong here? I am working on practice problems for my linear algebra course. The problem in question is as follows:
Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer.
$x_1+3x_2=f$
$cx_1+dx_2=g$
My first idea was to simply solve for d and c in terms of f and g:
$x_1+3x_2=f \implies \frac{x_1}{f}+\frac{3x_2}{f}=1$
$cx_1+dx_2=g \implies \frac{cx_1}{g}+\frac{dx_2}{g}=1$
Then these can be set equal to each other:
$\frac{x_1}{f}+\frac{3x_2}{f}=\frac{cx_1}{g}+\frac{dx_2}{g}$
$\implies \frac{1}{f}=\frac{c}{g}, \frac{3}{f}=\frac{d}{g}$
$\implies \frac{g}{f}=c, \frac{3g}{f}=d$
$\therefore d=3c$
I believed this answer would hold true, at least for nonzero values of f and g. However, row reduction on the system shows the exact opposite:
$ \begin{array}{lcr} \mbox{1} & 3 & f \\ \mbox{c} & d & g \\ \end{array}\ $
$\implies$
$ \begin{array}{lcr} \mbox{1} & 3 & f \\ \mbox{0} & d-3c & g-fc \\ \end{array}\ $
This matrix suggests that $g\neq3c$ except when f and g are zero. Can anyone tell me what is wrong with my first approach? Looking at it, I can tell that my answer is wrong (unless $g=cf$), but I don't see what is wrong with the algebra.
You can't jump from$$\frac{x_1}f+\frac{3x_2}f=\frac{cx_1}g+\frac{dx_2}g\tag1$$to$$\frac1f=\frac cg\text{ and }\frac3f=\frac dg.$$You could if the equality $(1)$ was true for every $x_1$ and every $x_2$, but you are assuming it only for some $x_1$ and some $x_2$.