Dear Uncle Colin,

I’ve got a line with equation $10y+36x=16.5$. That equation has no negative numbers in it, yet its gradient is apparently negative. I don’t understand why.

-- Silly Line, Only Positive Equation


It looks like we’re in misconception-land! In fact, you can write the equation of any line without negative signs by simply rearranging the equation so that everything is positive. I don’t recommend it, though: it’s generally a better idea to rearrange it into a form that gives you the gradient directly. In particular, the trick here is to isolate $y$ as follows:

$10y + 36x = 16.5$ – now subtract $36x$ from both sides:

$10y = -36x + 16.5$ – ((I’ve written it in a slightly unnatural-looking “$x$s first” pattern, which is normal for lines)) then divide by 10:

$y = - 3.6x + 1.65$ – ((Don’t forget to divide the constant by 10, too!)) Now do you see a -3.6?

When written in $y=mx+c$ form ((shut it, @srcav)), the number in front of the $x$ gives the gradient of the line. You could also get the gradient by differentiating, but that’s a story for another day.

-- Uncle Colin