I have a question regarding the terminology of the types of equations.

I understand that "Linear equations" are named like that because of you plot the solution pairs of (x,y), you will get a line on the Cartesian coordinate plan.

However I do not understand why equations with only one variable are still called linear equations. The solution to that type of equations do not form a line on the graph. Why are they still called "linear equations"?

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    $\begingroup$ $ax+by+c=0$ with at least one of $a,b \ne 0$ describes a line in the 2D plane. If $b=0$ the line is vertical, yet still a line. $\endgroup$ – dxiv Jul 23 '17 at 4:34
  • $\begingroup$ The definition of a linear equation does not refer to lines in a plane. A line in a plane results from graphing the special case of a linear equation in two variables. $\endgroup$ – N. F. Taussig Jul 23 '17 at 8:38
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    $\begingroup$ @N.F.Taussig Thanks, so linear equations can have as many variables as they want, but they should be variables to the power of 1 and it should not have variables multiplied. Is it correct? $\endgroup$ – yoyo_fun Jul 23 '17 at 9:06
  • $\begingroup$ That is correct. $\endgroup$ – N. F. Taussig Jul 23 '17 at 9:07

If you graph $3x-5=0$ on the $xy-$plane, you get the line that contains the points $(5/3,y)$ - $y$ can be any value. It is a vertical line.

If you graph $3x-5=0$ on the $x-$line (so the number line with $x$ as a variable), then it just forms the single point $(5/3)$.

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    $\begingroup$ But if we plot the solution to a two variable equation on the XYZ plane then we get a plane (if I am not mistaken), because Z can be any value. By this logic two variable equation can be called "planar equations". $\endgroup$ – yoyo_fun Jul 23 '17 at 4:30
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    $\begingroup$ @yoyo_fun Sure, and they would be if you were talking about the equation in space, not on a plane. $3x+y=5$ is a linear equation on the plane, a planar equation in space, etc. It all depends on what the reference space is (for example, as I said, the equation $3x-5=0$ makes sense in a linear reference space - the number line - and there it is just a "point equation"). $\endgroup$ – Carl Schildkraut Jul 23 '17 at 4:32
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    $\begingroup$ @yoyo_fun You are basically answering your own question. An equation will represent different geometric objects in different contexts, and both you and Carl have provided examples of this concept. $\endgroup$ – angryavian Jul 23 '17 at 4:34
  • $\begingroup$ @CarlSchidkraut so the name "linear equations" is more a convention. And what would happen if we would plot a one-variable equation on the three-dimensional xyz plane. I am a little bit confused about that. $\endgroup$ – yoyo_fun Jul 23 '17 at 4:35
  • $\begingroup$ @yoyo_fun Any equation that looks like $ax+by+cz=d$ for some reals $a,b,c,d$ (some of them may be zero, but $a,b,c$ may not all be $0$ - otherwise it is called "degenerate") will be a plane in three-dimensional space. $\endgroup$ – Carl Schildkraut Jul 23 '17 at 4:37

In general, a linear equation is of the form $$c_1x_1+c_2x_2+...+c_nx_n=b,$$ where $c_i\neq0$ for all $i$ and $b$ is a constant. The solution is a $(n-1)$-dimensional hyperplane in $n$-dimensional space.


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