What is the Geometric Meaning of Adding Equations Side-by-Side?

[Taturana]

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]

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.

 

[Taturana]

Thank you for the reply.

But, what happens to the lines geometrically?

 

[Redbelly98]

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]

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!

 

[Redbelly98]

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​

Add them together to get

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]

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

 

[Mentallic]

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]

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. And if you divide them such as to make the coefficients of x the same you get the other bisector?

 

[Redbelly98]

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]

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]

So mentallic was right to begin with?

He was right for that one example, but it doesn't hold in general.

If I divide one equation by a number it's still the same line.

Yes, see my post #6 ...

It is rather late at night.

... tomorrow.