Adding equations side-by-side

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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
 

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  • #2
arildno
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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.
 
  • #3
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Thank you for the reply.

But, what happens to the lines geometrically?
 
  • #4
Redbelly98
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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.
 
  • #5
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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  • #6
Redbelly98
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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​
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.
 
  • #7
Mentallic
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Hmm yes you're right. I'm going to put a bit more thought into this one.
 
  • #8
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).
 
  • #9
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?
 
  • #10
Redbelly98
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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.
 
  • #11
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.
 
  • #12
Redbelly98
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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.
 

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