Two-Step Equations
A two-step equation is exactly what it sounds like — an equation you can solve in two moves. Equations like or have a variable that got multiplied by something and then had something added or subtracted. Your job is to reverse both of those, one at a time.
The order matters, and it's the opposite of the order of operations: deal with the addition or subtraction first, then the multiplication or division. Get that order down and every two-step equation is the same 20 seconds of work.
The two steps
Step 1: move the constant. If the equation adds a number, subtract it from both sides. If it subtracts a number, add it to both sides. After this step the variable term is alone on its side: something like .
Step 2: divide by the coefficient. The coefficient is the number attached to the variable by multiplication. Divide both sides by it (or, if the variable is divided by a number, multiply both sides instead). Now the variable is alone, and whatever is on the other side is your answer.
Whatever you do to one side, you must do to the other — that's the whole game. An equation is a balance, and both moves above keep it balanced.
Why the constant goes first
In , the expression was built by multiplying by first, then adding . To take it apart you work in reverse — last thing built, first thing removed. The went on last, so it comes off first.
If you divide first instead, it still works — but every term gets divided, fractions show up early, and that's where most mistakes happen. Constant first keeps the numbers whole for as long as possible.
Watch the signs
The classic trap is a negative coefficient. In , after subtracting from both sides you get — and you must divide by , not . The answer is . Whenever your last step divides by a negative, double-check the sign of your answer by plugging it back in.
Worked examples
Example 1: addition, then division
Solve .
Answer:
Example 2: subtraction, then division
Solve .
Answer:
Example 3: a negative coefficient
Solve .
Answer:
Example 4: the variable is divided
Solve .
Answer:
Try one yourself
Common questions
How do I know which number to move first?
Move the constant — the plain number being added or subtracted — first. Then divide by the coefficient. It's the reverse of the order of operations, because you're taking the expression apart instead of building it.
What if the answer isn't a whole number?
That's fine. If you end with something like , dividing gives . A fraction can be a correct answer — leave it reduced, or as a decimal if the problem uses decimals.
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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