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

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Discussion Overview

The discussion revolves around the geometric meaning of adding two linear equations side-by-side. Participants explore the implications of this operation in terms of the resulting line and its relationship to the original lines, considering both specific examples and general cases.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks about the geometric meaning of adding two equations that describe lines.
  • Another participant suggests that adding the equations results in a new line that passes through the intersection point of the original lines.
  • One participant claims the resultant line represents the axis of symmetry of the two original lines, but later retracts this statement, indicating it is not universally applicable.
  • Another participant provides a counterexample to the symmetry claim, showing that different equations can yield different resultant lines.
  • There is a discussion about how the slopes of the original lines affect the gradient of the resultant line, with one participant noting that extreme differences in gradients can influence the outcome.
  • Some participants discuss the effect of manipulating the equations, such as dividing by coefficients to achieve certain relationships between the lines.
  • There is a recognition that while one example may support a claim, it does not necessarily hold true in general cases.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the geometric implications of adding the equations. There are competing views regarding the nature of the resultant line and its relationship to the original lines, with some participants acknowledging the limitations of their claims.

Contextual Notes

Limitations include the dependence on specific examples and the varying interpretations of the geometric meaning of the resultant line. The discussion highlights the complexity of the relationships between the lines based on their coefficients and slopes.

Taturana
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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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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.
 
Thank you for the reply.

But, what happens to the lines geometrically?
 
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.
 
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!
 
Last edited:
Mentallic said:
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.
 
Hmm yes you're right. I'm going to put a bit more thought into this one.
 
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).
 
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. :rolleyes: And if you divide them such as to make the coefficients of x the same you get the other bisector?
 
  • #10
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
Redbelly98 said:
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
epenguin said:
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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