Algebra Question: order of calculation

I watched an instructor write this question 3(x-7)=6, he then solved the Parentheses to get 3x-21=6, but here is where I was lost, he then added 21 on each side to get 3x=27, then divided each side by 3 to get x=9. Now, I always thought you you did everything in this order: Parentheses, Exponents, multiply/divide, then add/ subtract...but when I do this to this question, I got x-21=2, then x=23...can someone explain why he added/subtracted, before multiplied/divided, when trying to find the value of 'x'?

• nvm, I see what I did wrong...I forgot to divide the 21 :/ – Duh VillageIdiot Mar 29 '16 at 4:09
• The order you are referring to has nothing to do with what you are doing here. When modifying equations or, if you prefer, when simplifying/solving equations you must do the same operation on both sides - always. Dividing the initial equation $3(x-7) = 6$ by $3$ on both sides could be considered an even faster way of solving for the $x$. Consider also the (silly) example $a(x-1) -1 = -1$ where adding $1$ on both sides as the first operation gives the solution immediately. – Therkel Mar 29 '16 at 11:35
• You always follow the order of operations when evaluating an expression. When solving an equation (i.e. or unraveling an evaluation of an expression) we need to reverse the order of operations. – John Joy Mar 29 '16 at 13:15

To solve the equation $3(x - 7) = 6$, we must isolate $x$. There are various options.
Method 1: Your instructor's method. \begin{align*} 3(x - 7) & = 6\\ 3x - 21 & = 6 && \text{apply the distributive law}\\ 3x & = 27 && \text{add $21$ to each side of the equation}\\ x & = 9 && \text{divide each side of the equation by $3$} \end{align*}
Method 2: A corrected version of your attempt. \begin{align*} 3(x - 7) & = 6\\ 3x - 21 & = 6 && \text{apply the distributive law}\\ x - 7 & = 2 && \text{divide each side of the equation by $3$}\\ x & = 9 && \text{add $9$ to each side of the equation} \end{align*} As you realized, you failed to divide $21$ by $3$.
Method 3: A simpler method that eliminates the need to apply the distributive law. \begin{align*} 3(x - 7) & = 6\\ x - 7 & = 2 && \text{divide each side of the equation by $3$}\\ x & = 9 && \text{add $7$ to each side of the equation} \end{align*} If you compare methods 2 and 3, you can see that applying the distributive law first introduces an extra step and, with it, extra opportunities to make an error. Notice that $$\frac{3x - 21}{3} = \frac{3(x - 7)}{3} = x - 7$$ Note that division by $3$ is multiplication by $1/3$. With this in mind, we see that the reason you had to divide $21$ by $3$ is the distributive law $a(b - c) = ab - ac$. $$\frac{3x - 21}{3} = \frac{1}{3}(3x - 21) = \frac{1}{3}(3x) - \frac{1}{3}(21) = x - 7$$ with $a = 1/3$, $b = 3x$, and $c = 21$.
Check: If $x = 9$, then \begin{align*} 3(x - 7) & = 3(9 - 7) && \text{substitute $9$ for $x$}\\ & = 3(2) && \text{perform the operation in parentheses}\\ & = 6 && \text{multiply} \end{align*} so the solution $x = 9$ is correct.