A very simple question.

If I have, for simplify, two equations that describes lines:

2x + 3y + 4 = 0
3x + 2y + 5 = 0

adding them side-by-side we get: 5x + 5y + 9 = 0

The question is: what happens if I add these two equations side-by-side? What's the meaning, what happens geometrically when I add these two equations?

Thank you

arildno
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Dearly Missed
The mathematical meaning is that if you add equal amounts to two other quantities, these being equal to each other as well, the resulting two quantities will also be equal.

But, what happens to the lines geometrically?

Redbelly98
Staff Emeritus
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It's just another line. All it really has in common with the original two lines is that it will pass through their intersection point. I don't think there's any more to it than that.

Mentallic
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The resultant line is the axis of symmetry of the two previous lines. You're in a way taking the average of both lines when you add them like that.

EDIT: WRONG!

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Redbelly98
Staff Emeritus
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The resultant line is the axis of symmetry of the two previous lines.
In this case, yes, but not in general.

Consider these two equations:
4x + 6y + 8 = 0
3x + 2y + 5 = 0​
7x + 8y + 13 = 0​
That is different than the line we obtained before, even though -- guess what?-- we started with the same two lines as before.

Mentallic
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Hmm yes you're right. I'm going to put a bit more thought into this one.

Mentallic
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Wait, of course it's not! How silly of me!
If one considers an extreme example of a line with a small gradient, and one with a very large gradient, the resultant line will too have a nearly as large gradient (but definitely not enough to become an approx 1/-1 gradient).

epenguin
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But if you divide one of the equations by a number such as to make the coefficients of y the same you get that result I think. :uhh: And if you divide them such as to make the coefficients of x the same you get the other bisector?

Redbelly98
Staff Emeritus
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But the original equations had different coef's for x, and different coef's for y ... yet we got the bisector.

In the OP's example, the slopes of the two lines were reciprocals of each other, so the bisectors clearly should have slopes of ±1.

epenguin
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But the original equations had different coef's for x, and different coef's for y ... yet we got the bisector.

So mentallic was right to begin with? If I divide one equation by a number it's still the same line.
It is rather late at night.

Redbelly98
Staff Emeritus