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Suppose I have a line $Ax+By+C=0 $ and another line $ \alpha x +\beta y +\gamma=0$ and I add the two equations, what does the new equation represent geometrically?

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marked as duplicate by Jam, Aqua, YuiTo Cheng, cmk, Adrian Keister Jul 19 at 19:43

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The short answer to your question is that there isn't a single answer to your question. I'll try to explain below.

Suppose that your two equations are $$Ax+By+C=0$$ and $$\alpha x+\beta y+\gamma=0.$$

Let's assume that these equations define lines, i.e., (either $A\not=0$ or $B\not=0$) and (either $\alpha\not=0$ or $\beta\not=0$). Let's also assume that their sum defines a line, i.e., either $A\not=-\alpha$ or $B\not=-\beta$.

In this case, we can observe that the sum of these two equations defines a line which passes through the intersection of the first two lines (since plugging in that point results in $0+0=0$).

The problem with a more geometric answer is that the equation for a line is not unique. In particular, for any $\lambda\not=0$, $$ \alpha\lambda x+\beta\lambda y+\gamma\lambda=0 $$ defines the same line as the second equation. When you add this equation to the first equation, you get $$ (A+\alpha\lambda)x+(B+\beta\lambda)y+(C+\gamma\lambda)=0. $$ As $\lambda$ varies, this defines an entire family of lines (also called a pencil) that pass through the intersection point of the first pair of lines. So, even though the geometry hasn't changed, the actual line you get has changed. Therefore, there is no unique answer to your question geometrically.

Some special cases: If the two lines are the same line, then the sum is the same line again. If the two lines are parallel (but not equal), the result is a new line parallel to the first pair (but not equal). The geometric issue above also shows up in this second special case where scaling the equation of one of the parallel lines results in a whole family of parallel lines.

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If you add the two equations together, then you get

$$(A+\alpha)x + (B+\beta)y + (C + \gamma) = 0$$

We can rename the summed coefficients ($A+\alpha = a$ and so on) without loss of generality:

$$ax + by + c = 0$$

Which we can see is another equation of a line

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When you add two equations of straight lines you will still end up with a straight line but with a different slope if your two initial lines were non-parallel.

One way to visualize this process is by thinking about what happens to the individual points of these two lines and resulting line with same x-coordinates.The corresponding point of the resulting line will have it's y-coordinate as the algebraic sum of y-coordinates of the corresponding points of the two initial lines. Connecting all these points will result in a straight line 'hovering over' the given two lines in the graph plotted.

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