Two-Step Equations
A two-step equation takes exactly two moves to solve. In , the variable was multiplied by and then had subtracted — so you reverse both operations, one at a time.
The order matters: handle the adding or subtracting first, then the multiplying or dividing. That's the reverse of the order of operations, because you're taking the expression apart instead of building it.
The two moves
Step 1: move the constant. If a number is added to the variable term, subtract it from both sides; if it's subtracted, add it. After this step the variable term stands alone: something like .
Step 2: deal with the coefficient. If the variable is multiplied by a number, divide both sides by it. If the variable is divided by a number, multiply both sides instead. Now the variable is alone, and the other side is your answer.
Why the constant moves first
In the expression was built by multiplying first and subtracting second. Taking it apart runs in reverse: the last operation applied comes off first, so the moves before the .
Moving the constant first also keeps the numbers whole longer. Divide first and every term turns into a fraction — that's where most avoidable mistakes happen.
Negatives to watch
If the variable term is negative, like in , the last step divides by the whole coefficient, sign included. Subtracting gives , and dividing by gives . Whenever that final division uses a negative, plug the answer back in to confirm the sign.
Worked examples
Example 1: subtract, then divide
Solve .
Answer:
Example 2: add, then divide
Solve .
Answer:
Example 3: the variable is divided
Solve .
Answer:
Try one yourself
Common questions
Which number do I move first?
The constant — the plain number added to or subtracted from the variable term. Then handle the coefficient. That order is the reverse of the order of operations, because you're taking the expression apart.
What if the equation starts with the constant, like ?
Same plan. Subtract from both sides to get , then divide by to get . The variable term is , so the division has to carry the negative sign.
What if the answer is negative or a fraction?
Both happen all the time and both are fine. Solving gives ; solving gives . Leave fractions reduced.
How do I check my answer?
Substitute it back into the original equation. If both sides come out equal, it's right. This takes ten seconds and catches nearly every sign mistake.
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