Do I Substitute Values into an Equation Before or After Simplification of the Expression and Why?

Question

Considering the equation: \begin{align} y&=\dfrac{6x+x^2}{x} &\text{(before simplification)}\\ y&=6+x &\text{(after simplification).} \end{align}

• If Before Simplification of the expression, $x = 0$ is substituted, $y = undefined$
• But after Simlfication, $x = 0$ is substituted, $y = 6$

Why is this so? Or am I getting something wrong?

Answer 1

This is because you did the simplification almost correct. The actual correct simplification would be $$y = x + 6, \quad x \neq 0$$ because that's what captures all information in the original equation. If you forget the $x \neq 0$, you lose that piece of information, and that leads to your confusion.

Answer 2

$$y = \frac{6x+x^2}{x} = \frac{x(6+x)}{x}$$ Now, you cannot cancel out the $x$ if $x = 0$ because division by zero is undefined. Your simplification is valid for any $x \neq 0$ $$ y = 6+x ~~~\forall ~x \neq 0 $$ Another thing to note here is that the domain of $y(x)$ is $\mathbb{R} - \{0\} $ which means that $y$ is not defined at $x=0$. Therefore, the graph of $y(x)$ will be a straight line ($y=x+6$) missing the point $(0,6)$. However, if you plot $y=x+6$ , the graph will be a straight line including $(0,6)$. Apart from the point $(0,6)$, both the graphs are identical.