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

• Taturana
In summary, adding two equations describing lines together results in a new line that passes through the intersection of the original two lines. However, this resultant line is not always the axis of symmetry of the original lines, as it depends on the slopes of the original lines. If the slopes are reciprocals of each other, then the resultant line will be the axis of symmetry, but in general, this is not the case.
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

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?

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

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.

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.

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.

## What is the purpose of adding equations side-by-side?

The purpose of adding equations side-by-side is to combine two or more equations to solve a problem or to find a common solution. This method is often used in algebra and other mathematical fields to simplify complex equations.

## Can any type of equation be added side-by-side?

Yes, any type of equation can be added side-by-side as long as they have the same variables and can be solved using the same methods. However, the resulting equation may become more complex and difficult to solve.

## What are the steps for adding equations side-by-side?

The steps for adding equations side-by-side are as follows:
1. Identify the common variables in both equations.
2. Rearrange the equations so that the variables line up.
3. Add the coefficients of the variables together.
4. Simplify the equation by combining like terms.
5. Solve for the remaining variable, if necessary.

## Is there a limit to the number of equations that can be added side-by-side?

No, there is no limit to the number of equations that can be added side-by-side. However, as the number of equations increases, the complexity of the resulting equation also increases, making it more difficult to solve.

Adding equations side-by-side can help simplify complex equations, making them easier to solve. It can also help find solutions to problems that may have multiple variables or unknowns. This method can also be used to check the accuracy of solutions obtained through other methods.

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