# The slope of the ratio of two lines

Discussion in 'Geometry and Trigonometry' started by StockholmEd, Dec 22, 2022.

1. ### StockholmEd

Joined:
Dec 22, 2022
Messages:
3
0
Hi, folks!

I need to check that given two lines both of whose slope is negative, then dividing one line by the other yields a third line whose slope is positive. I tried tinkering around with the algebra, but I can't seem to prove it.

Basically given two lines, y1 = m1x + c1, and y2 = m2x + c2, and given that m1 and m2 are both < 0, I thought it would be easy to show that if I divide the first line by the second, then the result would itself be a line whose slope is > 0.

But I can't seem to work it out.

I also tried selecting two points on both lines, via m = (y2 - y1)/(x2 - x1) and simply dividing the coordinates and then trying to create the line formula, but I can't separate the slope of the resulting equations into a form that depends only on x (it keeps getting mixed-up with y and the the c1, c2 constants).

Am I trying something that makes no sense?

Thanks,

Ed

StockholmEd, Dec 22, 2022

2. ### Phrzby Phil

Joined:
Mar 8, 2022
Messages:
12
3
What do you mean by "dividing one line by the other" ?

If you mean forming the rational function (m1x + c1)/(m2x + c2), then in general you will not get a linear function, and therefore not a straight line, and therefore not a slope.

Example: graph (-x+5) / (-2x+7)

(BTW - why do you use c1 and c2 when conventional terminology is b1 and b2 ?)

Phrzby Phil, Dec 23, 2022

3. ### StockholmEd

Joined:
Dec 22, 2022
Messages:
3
0
Oh, thank you! I hadn't thought of just plugging it into some graphing software.

Very interesting.

(And I use c1/c2 because that's what was in maths books 40 years ago; had no idea b1/b2 was the convention now.)

StockholmEd, Dec 23, 2022
4. ### Phrzby Phil

Joined:
Mar 8, 2022
Messages:
12
3
So did you mean (m1x + c1)/(m2x + c2) ?

Just curious - what is the actual problem, or was this just a curiosity - also a legitimate pursuit.

Also, only books I've ever seen say y = mx+b, where m = slope and b = y-intercept.

Phrzby Phil, Dec 23, 2022
5. ### StockholmEd

Joined:
Dec 22, 2022
Messages:
3
0
Yes indeed: I did mean (m1x + c1)/(m2x + c2).

I'm a programmer and trying to calculate the probability of a function's being updated based on its size.

"Also, only books I've ever seen say y = mx+b, where m = slope and b = y-intercept."- Ah. I'm sure I'm misremembering then. It's been an age since I last read this stuff.

StockholmEd, Dec 23, 2022