I apologise if this is too simple a level for this board.
The question is:
"Given the point $P(x_1,y_1)$ and the line $l$ with the equation $ax + by = c$:
Write an equation of the line $j$ that passes through $P$ and is perpendicular to $l$"
I am struggling to figure out this answer. I start with the given equation $l$ and reorient it into the form $y=ax+c$, essentially because I find that form easier to read for meaning. This, after canceling for $b$ in the given equation $l$ leaves me with:
$$y=\frac abx +\frac cb$$
Then, going by perpendicularity, I take the opposite reciprocal of the slope and end up with:
$$y_1=\frac bax_1 + …$$
And, like that, I suddenly realise that I have no idea what to input for $c$ here. Thinking it through, the $c$ is the y-intercept. Sketching out two lines on a grid, it seems like c should be equal to the initial y-intercept added to the new y-intercept but I don't figure how to figure this. Initially, I just reversed the slope for $-a$ and swapped that in but I feel that the $b$ needs to be accounted for as that would influence the slope. Really, this question has made me realise I don't understand the meaning of the various equations of the line fully, which is really positive, if currently frustrating.
In addition, the book yields the answer $bx-ay=bx_1-ay_1$
This answer seems to confirm I'm reasonably lost. I don't see, from here, why the $a$ and $b$ have switched on both sides between $x$ and $y$. Or why the general $x$ and $y$ of the original equation are set equal to the new line's points. I feel like I may be approaching this entirely wrong, and will continue trying. Any assistance would be greatly appreciated.
Also, again, I apologise if this is too low a level for this stackexchange. I am working on improving my math skills as an adult and live in an area where I have been unable to get math tutoring in English so I am working as I can by myself. Your help would be very kindly appreciated.
Best Answer
You aren't lost, you just quit too early.
$y_1=\frac bax_1+...$ gives you the value of the unknown intercept, $...=y_1-\frac bax_1$.
Hence,$$y=\frac bax+y_1-\frac bax_1.$$