# Defining linear equation without plotting y against x

Without plotting $y$ against $x$ how could you tell the below were linear equations $$y = 100 - \frac{9}{x}$$ and $$y = 2x^2 -10x.$$

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One quick way is that linear equations have a degree of one (or, technically, zero). So, the x variable can only be raised to the power of one, no higher or lower. No x^2 or x^3 anywhere in the equation.

The first one has a x in the denominator, that's basically multiplying by an x to the power of -1. The second one has an x to the power of 2.

It depends on how your homework wants you to prove it, though.

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Using the definition of linear equation. A linear equation in variables $x$ and $y$ is an equation of the form: $$Ax+By=C,$$ for fixed constants $A$, $B$ and $C$ (with $A^2+B^2>0$). Do any of the equations you listed have this form?

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I almost wrote that, but $AB\neq0$ means that both $A$ and $B$ must be nonzero... but that's not the case, as $y=1$ is a valid linear equation. – dls Dec 13 '11 at 20:07

Well, the first one certainly isn't a linear equation because you have $1/x$ term in it. Linear always means you only have $x$ appearing in the equation. So definitely, the second one is a linear equation.

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There is a "not" missing in the last sentence. – André Nicolas Dec 13 '11 at 19:29