Given the equation:

$$-\dfrac{3}{x} + y = 10$$

Is this equation linear? Yes or no? Please explain.

I have tried 7 other problems like this and easily figured it out, since this is written differently (a negative fraction and the $y$ isn't by itself).

I don't know what to do.

  • $\begingroup$ What are your thoughts? What is the definition of linear? Have you tried rearranging it to a more familiar form? $\endgroup$ Jul 27, 2013 at 23:20
  • $\begingroup$ Can you think of an equivalent expression that has the $y$ by itself? What if you added $3/x$ to both sides of the equation? $\endgroup$ Jul 27, 2013 at 23:22
  • $\begingroup$ Arranged in or extending along a straight or nearly straight line: "linear arrangements" is the definition. my thoughts are i would like for someone to tell me what to do in the case of this problem. since it is so differently written then others. $\endgroup$
    – user87954
    Jul 27, 2013 at 23:22
  • 1
    $\begingroup$ user87954: Is that $\dfrac{-3}{x+y}$ or $\dfrac{-3}{x}+y$ on the left side of the equation? (I would assume the second one, but want to be sure.) $\endgroup$ Jul 27, 2013 at 23:23
  • $\begingroup$ I agree with @Jonas . Your comment "the "$y$" isn't by itself" suggests that, in fact, the left hand side should be the first one suggested by Jonas in his comment directly above. If so, then the equation is linear ... $\endgroup$ Jul 27, 2013 at 23:30

4 Answers 4


If the equation were linear, then its graph (in the $xy$-plane) would be a straight line. However, check that the following points indeed lie on the graph:




Can there be a straight line passing through these three points?

I hope this helps! (Of course, there's a formal proof that the equation isn't linear given in the nice answer by T. Bongers above. However, when you see an equation and you think it's linear (or not), then you should note that the graph must be a straight line. That's the intuition behind linear equations. It's worth plotting the entire graph to see what it looks like but just plotting the three points above should show you that the equation is not linear.)


The following image is provided by User:ali (https://math.stackexchange.com/users/78741/ali). I think it's a fantastic addition to my answer and credit should go to him for it (thanks!). (You can clearly see here that the equation isn't linear. Exercise: Can you confirm visually that the three points above lie on this graph?)

enter image description here


Upon rearrangement, we find

$$-3 + xy - 10x = 0$$

This is not linear, since linear equations can always be put in the form

$$ax + by + c = 0$$

The $xy$ means we can't.


For Lienar equations in two or more than two variables,

In order to be a linear equation, the powers of $x_1, \dots , x_n$ must be equal to $1$, not be bigger than $1$.

The general linear equation in n variable is the following form;

$$ a_1x_1+\cdots + a_nx_n=c$$ $a_i$'s any number. And $c$ is constant.

As for your equation,


This equation consists of two variables $x$ any $y$

The powers of $x$ are all $1$. And the power of $y$ is $1$.

All is okay so far. However,

since $\exists$ $xy$ in the equation, the equation is not linear.

  • $\begingroup$ Nice answer, B11b! $\endgroup$ Jul 28, 2013 at 5:42

Here is the general form for a linear equation, so you can compare

$$y := m x + b\quad m,b\in \mathbb{R}. $$


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