TITLE: Point by point, the transformation $T(x, y) = (4x - y, 3x -2y)$ sends the line $x + 2y = 6$ onto an image line. What is the slope of the image?
QUESTION [5 upvotes]: The question is as follows:

Point by point, the transformation $T(x, y) = (4x - y, 3x -2y)$ sends the line $x + 2y = 6$ onto an image line. What is the slope of the image? 

I don't understand what this question basically. How is it that the transformation is able to send a line into an "image line" (a term that I am not even familiarized with)? Any help will be greatly appreciated.

REPLY [3 votes]: $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$.
$T(x,y) \mapsto (x',y')$,
$x'=4x-y$;  $y'=3x-2y.$
Solving for $x,y$ in terms of $x',y'$.
$2x'-y'=5x$;  and $ 3x' -4y' =5y$.
The line $x+2y = 6$ is transformed:
$5x +10y =30 \rightarrow$
$(2x'-y' ) + 2 (3x'-4y') =30.$
$8x' - 9y' = 30.$
Slope of the image line:  $m' = 8/9.$