Two-step equations

Worked example

2.2.w1 Multiply-then-add: $$\begin{aligned} &2x+3=11 \\ &\xrightarrow{\,-3\,}\; 2x=8 \\ &\xrightarrow{\,\div 2\,}\; x=4 \\ &\text{Check: } 2(4)+3=8+3=11 \end{aligned}$$

2.2.w2 Division type: $$\begin{aligned} &\frac{x}{3}-2=4 \\ &\xrightarrow{\,+2\,}\; \frac{x}{3}=6 \\ &\xrightarrow{\,\times 3\,}\; x=18 \\ &\text{Check: } \frac{18}{3}-2=6-2=4 \end{aligned}$$

2.2.w3 Dividing by a negative: $$\begin{aligned} &8-2x=14 \\ &\xrightarrow{\,-8\,}\; -2x=6 \\ &\xrightarrow{\,\div(-2)\,}\; x=-3 \\ &\text{Check: } 8-2(-3)=8+6=14 \end{aligned}$$

That third one is worth slowing down for, because it's where a sign quietly goes missing. After you subtract 8, the left side is −2x, not 2x. The minus belongs to the term. So you divide both sides by −2, and 6 ÷ (−2) is −3.

The check is where a slip would have shown up: 8 − 2(−3) means subtracting a negative, which adds, giving 8 + 6 = 14. A negative answer here isn't a warning sign. It's a real point on the line, and the check confirms it.

2.2.w4 Here's a clean one to get the rhythm back before you practice. The answer comes out to 0, which surprises people, but zero balances the scale just fine: $$\begin{aligned} &3x+6=6 \\ &\xrightarrow{\,-6\,}\; 3x=0 \\ &\xrightarrow{\,\div 3\,}\; x=0 \\ &\text{Check: } 3(0)+6=6 \end{aligned}$$

One slip is easy to make on the very first move, now that the layers are stacked: undoing in the wrong order. On 2x + 3 = 11, dividing by 2 before subtracting the 3 forces you to divide the 3 as well, which drags in a fraction and makes the numbers messy. The getting-dressed picture is the fix. Whatever went on last comes off first, so the +3 goes before the ×2.