# Linear Algebra: is it acceptable to make any equation containing $x$ equal $0$?

Is it acceptable to make any equation containing $x$ equal $0$?

For example:

$$\frac{2}{3x + 1}$$

Is it acceptable to make this equation equal zero?

$$\frac{2}{3x + 1} = 0$$

I'm slightly confused, In my text book it states that the y asymptote of $\frac{2}{3x + 1}$ is $0$, does this mean it is incorrect to ever make this equation equal zero?

Regardless, when is it ok to make an equation equal $0$ and when is it not ok to make an equation equal $0$? (Obviously assuming the equation has an unknown variable in it, $x$)

• What do you mean by "acceptable"? There's nothing stopping you in consider (or not) an equation $f(x)=0$. But whether that's useful or not totally depends on the context. What exactly are you trying to achieve? Commented Mar 1, 2017 at 8:54
• Actually, before you add "$=0$", this is not even an equation. Hiwever, if you have a numerical expression such as here, it is always syntactically correct to turn it into an equation by appending "$=0$" Commented Mar 1, 2017 at 8:54
• Just so you know a horizontal asymptote doesn't mean zero cannot be a solution consider $\frac{x}{x^2+1}$ where x is real, this function has an asymptote of $y=0$ yet also has an answer for x when set equal to zero. A horizontal asymptote tells you about the functions behavior when x becomes very large. Commented Mar 1, 2017 at 8:57

It is fine that you extend your term to an equation.

If the resulting equation has solutions for $x$ which fulfill the equation is not granted. It could have or could have not.

I'm slightly confused, In my text book it states that the $y$ asymptote of $2 / (3x + 1)$ is $0$, does this mean it is incorrect to ever make this equation equal zero?

No. It might give a hint that the equation has no solution. Meaning there is no value for $x$ which satisfies the equation (makes it a true statement).

Regardless, when is it ok to make an equation equal $0$ and when is it not ok to make an equation equal $0$? (Obviously assuming the equation has an unknown variable in it, $x$)

See above. You are allowed to formulate equations.

In case your are interested in solutions for your example: $$0 = \frac{2}{3x+1}$$ For any real number $x$ the right hand side will be either positive or negative, but not zero. So there is no solution within the set of real numbers.