Multi-Step Equations
A multi-step equation is a two-step equation with some cleanup in front. Equations like or look longer, but they aren't harder — the side with the variable just isn't simplified yet. Once you tidy it up, you're back to a two-step problem you already know how to finish.
So the whole strategy is: simplify first, solve second. Distribute any parentheses, combine any like terms, and only then start moving numbers across the equals sign.
The game plan
Step 1: distribute. If there are parentheses with a number in front, multiply it through every term inside.
Step 2: combine like terms. On each side separately, add up the variable terms and add up the plain numbers. becomes .
Step 3: finish like a two-step equation. Move the constant to the other side, then divide by the coefficient.
Notice that steps 1 and 2 never touch the other side of the equation — you're just rewriting one side in a simpler form. Only in step 3 do you start doing matching moves to both sides.
Combining like terms first
Like terms have the exact same variable part: and are like terms, but and are not. In , the like terms are and — and the sign in front of a term travels with it, so they combine to , not .
Combining before solving matters because each side of the equation should be as short as possible before you start moving things. Trying to subtract terms across the equals sign while the side is still messy is where lost terms and sign errors come from.
Distributing without losing a term
In , the multiplies both the and the : . The most common mistake is multiplying only the first term and writing .
Also watch negatives: is , because the times gives . Whenever you distribute a negative, every sign inside the parentheses flips.
Worked examples
Example 1: combine like terms, then solve
Solve .
Answer:
Example 2: distribute, then solve
Solve .
Answer:
Example 3: distribute and combine
Solve .
Answer:
Try one yourself
Common questions
How do I know what to do first?
Simplify before you solve. If there are parentheses, distribute. If there are like terms on one side, combine them. Only when each side is as simple as it can get do you start moving terms across the equals sign.
Can I combine terms from opposite sides of the equals sign?
Not directly. Combining like terms only works within one side. To deal with a term on the other side, add or subtract it from both sides — that's a solving move, not a simplifying move.
What if there's a negative in front of the parentheses?
Distribute it to every term and flip every sign inside: becomes . Write the distributed line out fully before doing anything else — skipping it is the top source of errors.
How do I check my answer?
Plug it into the original equation, not the simplified one. If both sides match, you're right — and if they don't, the check tells you a mistake happened somewhere in your steps.
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