# Order of operations when balancing equation

This is a trivial question, yet I have only really thought about it today and would like some insight.

To find the $x$-intercept of the following function:

$$f(x) = 2 \sin x + 1$$

We set $f(x) = 0$, subtract $1$ from both sides, divide by $2$ and get the answer: $\sin x= -\frac 12$.

But if we divide by $2$ first, then subtract $1$, we get the answer: $\sin x = -1$.

How do we know which operation to "undo" first in order to obtain the correct result?

• How do you get sinx=1 by dividing by 2 first? You need to divide the 1 by 2 as well – Quality Mar 23 '15 at 19:12
• You're right. I'm an idiot! Thanks – Jordan Mar 23 '15 at 19:23

If you divide first by $2$ we have $$2\sin x + 1 = 0 \iff \sin x + \frac 12 = 0 \iff \sin x = -\frac 12$$
So either operation can be the first to be applied. What you failed to do was divide $1$ by 2.