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Even though this seems to be an absurd question I would like to know why we can't add different variables together?What keeps us from not adding unlike variables together?

For example if we have an system of linear equation as:

  1. $1x_1 + 2x_2 = 3$
  2. $2x_1 + 4x_2 = 6$

we are obligated to add $1x_1$ and $2x_1$ together instead of $1x_1$ and $4x_2$.

Could anyone tell the reason of adding two same variables together instead of different variables.

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closed as unclear what you're asking by lulu, Hans Lundmark, user91500, Daniel W. Farlow, Arnaud D. Mar 30 '17 at 10:06

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    $\begingroup$ No idea what you are asking. Can you try to clarify? $\endgroup$ – lulu Mar 29 '17 at 18:46
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    $\begingroup$ There is no reason you can't add $1x_1$ and $4x_2$, but the problem is that you don't know what that will equal. $\endgroup$ – ChocolateAndCheese Mar 29 '17 at 18:50
  • $\begingroup$ Well, if you really want to add them, you get $1x_1+4x_2=-2x_2-2x_1-3$ (using substitution), but that's not really useful, is it? We add $1x_2$ and $2x_2$, because the system gives us that it equals 3. Same for $2x_1$ and $4x_2$. We are not given (directly) what it means it we add $1x_1$ and 4x_1$, so there is no point in adding those. $\endgroup$ – Sha Vuklia Mar 29 '17 at 19:07
  • $\begingroup$ @ShaVuklia:Do you mean to say we can't visualize of adding two equations on a graph? $\endgroup$ – justin Mar 29 '17 at 19:12
  • $\begingroup$ @justin Adding two equation would mean that you add everything on the left side together and everything on the right side together. So you would get: $3x_1+6x_2=9$, but that's not really going to help you further. If you add the two equations, you simply get a new equation, and you can graph it, if you want. I would suggest you try to let go of the visualisation as a means of understanding the system - at least for now - and focus on the algebraic properties instead. (That's way easier) $\endgroup$ – Sha Vuklia Mar 29 '17 at 19:17
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When you add the two equations, you get a sum of four terms on the left. It's common to simplify by combining the coefficients with similar variables, but by no means necessary. You're welcome to write $(1x_1+2x_2)+(2x_1+4x_2)$ and leave it like that. Or write $(1x_1+4x_2)+(2x_2+2x_1)$. It's common to write $(1x_1+2x_1)+(2x_2+4x_2)$ because then we can easily observe that this is equal to $3x_1+6x_2$, but writing the sum in that form is a simplification of the four-term sum. All four of those sums are equal. So, in some sense you're asking the wrong question - you mean to ask about simplifying sums.

So let's ask the right question: why do we simplify like that? Because $x_1$ and $x_2$ are different types of things. Imagine that $x_1$ are rocks and $x_2$ are flowers. If you have $1$ rock and $2$ flowers on Monday and $2$ rocks and $4$ flowers on Tuesday and someone asks you how many total rocks and flowers you had between the two days and you said $1+4=5$ rocks and $2+2=4$ flowers, you're wrong because you're failing to correctly identify objects as rocks vs flowers. Likewise, you have a certain amount of $x_1$ and a certain amount of $x_2$ and you're failing to draw this distinction.

One common objection to this explanation is that $x_1$ and $x_2$ are numbers, and therefore the same kind of thing. That's half right, but misleadingly so. $1$ kilometer and $1$ mile are both units of distance, but that's not the right sense of the phrase "kind of thing." If we redo the above example about distances walked, the exact same objection applies to the rocks and flowers case.

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  • $\begingroup$ :I have thought about this answer while posting this question.Could you tell graphically why we can't add unlike variables.More clearly what happens if add these variables together? $\endgroup$ – justin Mar 29 '17 at 18:57
  • $\begingroup$ @justin you can add $2x_1$ to $3x_2$. You get $2x_1+3x_2$, which happens to not be the same thing as $5x_1$. Does that part make sense to you? $\endgroup$ – Stella Biderman Mar 29 '17 at 18:58
  • $\begingroup$ @justin also, I have heavily edited and expanded my answer so check to make sure you read the most recent version $\endgroup$ – Stella Biderman Mar 29 '17 at 18:59
  • $\begingroup$ :That's true cause no one add an rock and a flower as you told.Could you tell what happens when we add two same variables graphically together? $\endgroup$ – justin Mar 29 '17 at 19:02
  • $\begingroup$ @justin I don't know what that means. Do you mean "draw a picture"? What kind of picture. "Adding things graphically" doesn't refer to anything in particular. $\endgroup$ – Stella Biderman Mar 29 '17 at 19:02
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When we say $2x_1$ the $x_1$ could be anything. It could be $7$. It could be $5,924,791.57$. Or anything else. So $2x_1$ could be anything. It could be $14$. It could be $11849583.14$. But its value will depend on whatever $x_1 $ is. If you know what $x_1$ is you know what $2x_1$ is.

And when we say $3x_2$ then $x_2$ could be anything. It could be $2$ and it could be $-4,723,206.14$. Or anything else. So $3x_2$ could be anything. It could be $6$ and it could be $-14169618.42$. But its value will depend on whatever $x_2$ is. If you know what $x_2$ is you know what $3x_2$ is

$2x_1 + 3x_2$ could be anything. It could be $20$ if $x_1 = 7$ and $x_2= 2$. It could be $-14169604.42$ if $x_1 =7$ and $x_2 = -4,723,206.14$. Or could be anything else. What it equals will depend on two different variables. If $x_1=7$ then $2x_1 + 3x_2$ could be wildly different values depending on what $x_2$ is. If what know what $x_1$ is you do NOT know what $2x_1 + 3x_2$ is because $x_2$ could vary wildly. If you know what $x_2$ is you do not know what $2x_1 + 3x_2$ is because $x_1$ could vary wildly.

We can not express $2x_1 + 3x_2$ in terms of a single variable because it is not dependent on only one variable. It's dependent on two variables each of which could have values completely independently of the other.

So the real question isn't why cant we add $2x_1 + 3x_2 = 5x_{somethingelse}$ but why CAN we add $2x_1 + 5x_1$ to get $7x_1$?

Well, because both $2x_1$ and $5x_1$ are dependent upon the same variable. If we know what $x_1$ is we have to know what both $2x_1$ is and $5x_1$ is. And if we know what both $2x_1$ and $5x_1$ is then we know what they both add up to.

We can never know what $2x_1 + 3x_2$ add up to if we only know one of the variables so we can't express a sum of both the variables in terms of just one.

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I think the answer is "since the axises $x_1$ and $x_2$ are orthogonal". So adding $x_1$ and $x_2$ would be like taking a couple of steps in the vertical and then in the horizontal direction but still you can't :

Add a step in the vertical direction by taking a step in the horizontal direction or vice-versa

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