➕ Algebra 1 · Equations
Solving Multi-Step Equations
Solve multi-step equations by distributing and combining like terms first, then undoing, including negatives and fractions, with memory tricks and practice.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- 8.EE.C.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
- A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
- A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.
By the end you can…
- Tidy an equation by distributing and combining like terms before you start undoing.
- Solve multi-step equations, including ones with negative numbers and fractions.
- Write an equation for a word problem and solve it.
- Check any solution by substituting it into the original equation.
Lesson (school style)
A multi-step equation is often a two-step equation in disguise: it may have parentheses, like terms scattered around, or fractions. The plan has three parts. Tidy each side on its own, undo what is left the way you did in two-step equations, then check in the original equation. In this lesson the variable stays on one side of the equals sign. The next lesson covers variables on both sides.
What makes an equation multi-step?
A multi-step equation takes more than two moves to get the variable alone. The extra moves usually come from parentheses, from like terms that need to be joined, or from fractions.
Look at 3(x + 2) + 2x = 31. You cannot undo anything yet, because x shows up in two places and one of them is trapped inside parentheses. The fix: tidy first.
Step 1: Tidy each side
Work on one side at a time, in this order.
- Distribute. If a number multiplies a group in parentheses, multiply it by every term inside: 3(x + 2) = 3x + 6.
- Combine like terms. Like terms have the same variable part, so 3x + 2x = 5x. A plain number only combines with another plain number.
For 3(x + 2) + 2x = 31, the left side becomes 3x + 6 + 2x, then 5x + 6. The equation is now 5x + 6 = 31.
Tidying only rewrites one side as a simpler expression with the same value. That is not a balance move, so you do not have to repeat it on the other side. If a side has no parentheses and no like terms to join, it is already tidy.
Step 2: Undo what is left
After tidying, you usually have an equation shaped like ax + b = c, which you already know how to solve. Undo the add or subtract first, then the multiply or divide. Every undo move is done to both sides (when you multiply or divide, use the same number on both sides, and never zero).
5x + 6 = 31 becomes 5x = 25 (subtract 6), then x = 5 (divide by 5). It does not matter which side the variable is on: 17 = 2x + 5 is solved with the same moves.
A shortcut that sometimes helps
If the parentheses are the entire side, like 6(x − 1) = 30, you can divide both sides by 6 first: x − 1 = 5, so x = 6. Distributing first gives the same answer: 6x − 6 = 30, then 6x = 36, then x = 6.
Dividing first works neatly when the parentheses make up the whole side. If another term sits outside the parentheses, like 6(x − 1) + 2 = 32, undo the + 2 first. That leaves 6(x − 1) = 30, and now you can divide by 6. Distributing first works too.
Watch the negatives
A number in front of parentheses includes its sign. For −2(x − 5), multiply −2 by x to get −2x, and −2 by −5 to get +10. So −2(x − 5) = −2x + 10.
A minus sign by itself in front of parentheses acts like multiplying by −1. It flips the sign of every term inside: 25 − (2x + 3) = 25 − 2x − 3. A plus sign in front changes nothing.
After tidying, you may land on a negative coefficient, like −2x = −12. Divide both sides by −2 to get x = 6. If you reach −x = 5, remember that −x means −1 · x, so divide both sides by −1 to get x = −5.
Fractions: two ways in
Clear the fractions. Multiply every term on both sides by a common denominator (the least one keeps the numbers small). For x/2 + x/3 = 10, multiply by 6 to get 3x + 2x = 60. Skipping even one term breaks the balance.
Use the reciprocal. If a fraction multiplies a whole group, like (2/3)(x − 6) = 8, multiply both sides by the reciprocal 3/2. That turns the 2/3 into 1, because (3/2)(2/3) = 1.
Either way you are multiplying both sides by the same nonzero number, so the solution does not change.
Always check in the original
Substitute your answer into the original equation, not the tidied one. Work out each side separately. If they match, you are done.
Why the original? If you slipped while tidying (say, you forgot to distribute to the second term), the tidied equation can pass its own check, but the original equation will not.
Key words
- Multi-step equation
- An equation that takes more than two moves to solve, usually because of parentheses, like terms to combine, or fractions.
- Distributive property
- a(b + c) = ab + ac. A number that multiplies a group multiplies every term in the group.
- Like terms
- Terms with the same variable part, such as 3x and 2x, or two plain numbers. 3x and 3 are not like terms.
- Equivalent equations
- Equations with exactly the same solution, like 5x + 6 = 31 and 5x = 25. Tidying and balance moves produce equivalent equations.
- Reciprocal
- For a nonzero number, the number you multiply it by to get 1. The reciprocal of 2/3 is 3/2. Zero has no reciprocal.
- Least common denominator (LCD)
- The smallest positive number that every denominator divides into evenly. For 1/2 and 1/3 it is 6.
Worked examples
Example 1. Solve 3(x + 2) + 2x = 31.
- Distribute the 3 to both terms in the parentheses: 3(x + 2) = 3x + 6. The equation is now 3x + 6 + 2x = 31.
- Combine the like terms 3x and 2x: 5x + 6 = 31.
- Undo the +6 by subtracting 6 from both sides: 5x = 25.
- Undo the ×5 by dividing both sides by 5: x = 5.
- Check in the original equation: 3(5 + 2) + 2(5) = 3(7) + 10 = 21 + 10 = 31. Both sides match.
Answer: Answer: x = 5
Example 2. Solve 6x + 4 − 2x − 9 = 23.
- There are no parentheses, so there is nothing to distribute. Each term keeps its sign: the x-terms are 6x and −2x, and the plain numbers are +4 and −9.
- Combine the x-terms: 6x − 2x = 4x. Combine the plain numbers: 4 − 9 = −5. The equation is 4x − 5 = 23.
- Add 5 to both sides: 4x = 28.
- Divide both sides by 4: x = 7.
- Check: 6(7) + 4 − 2(7) − 9 = 42 + 4 − 14 − 9 = 23. Both sides match.
Answer: Answer: x = 7
Example 3. Solve −2(x − 5) + 4 = 18.
- Distribute −2 with its sign: −2 · x = −2x and −2 · (−5) = +10. The equation is −2x + 10 + 4 = 18.
- Combine the plain numbers: −2x + 14 = 18.
- Subtract 14 from both sides: −2x = 4.
- Divide both sides by −2: x = 4 ÷ (−2) = −2.
- Check: −2(−2 − 5) + 4 = −2(−7) + 4 = 14 + 4 = 18. Both sides match.
Answer: Answer: x = −2
Example 4. Solve 25 − (2x + 3) = 10.
- The minus sign in front of the parentheses acts like multiplying by −1, so it flips every sign inside: 25 − 2x − 3 = 10.
- Combine the plain numbers 25 and −3: 22 − 2x = 10. The variable term is −2x, and its sign stays with it.
- Subtract 22 from both sides: −2x = −12.
- Divide both sides by −2: x = −12 ÷ (−2) = 6.
- Check: 25 − (2(6) + 3) = 25 − 15 = 10. Both sides match.
Answer: Answer: x = 6
Example 5. Solve x/2 + x/3 = 10.
- The denominators are 2 and 3, so the least common denominator is 6. Multiply every term on both sides by 6: 6 · (x/2) + 6 · (x/3) = 6 · 10.
- Simplify each term: 3x + 2x = 60.
- Combine like terms: 5x = 60.
- Divide both sides by 5: x = 12.
- Check: 12/2 + 12/3 = 6 + 4 = 10. Both sides match.
Answer: Answer: x = 12
Example 6. Solve (2/3)(x − 6) = 8.
- A fraction multiplies the whole group. Multiply both sides by the reciprocal 3/2: (3/2)(2/3)(x − 6) = (3/2)(8).
- The left side is 1 · (x − 6) and the right side is 24/2 = 12. So x − 6 = 12.
- Add 6 to both sides: x = 18.
- Check: (2/3)(18 − 6) = (2/3)(12) = 8. Both sides match.
Answer: Answer: x = 18
Example 7. Jordan and 2 friends each buy a movie ticket that costs t dollars and a 4-dollar snack. The group also pays one 3-dollar parking fee. The total is 51 dollars. How much does one ticket cost?
- Let t be the cost of one ticket in dollars.
- Each of the 3 people pays t + 4, and there is one 3-dollar fee on top: 3(t + 4) + 3 = 51.
- Distribute: 3t + 12 + 3 = 51, so 3t + 15 = 51.
- Subtract 15 from both sides: 3t = 36. Divide both sides by 3: t = 12.
- Check: 3(12 + 4) + 3 = 48 + 3 = 51. A ticket costs 12 dollars.
Answer: Answer: t = 12, so one ticket costs 12 dollars
In plain English
Some equations come in messy. They have parentheses, or the same kind of thing shows up in two places, or there are fractions. You cannot start undoing yet.
So you tidy up first. If a number sits in front of parentheses, share it with everything inside. If you see the same kind of term twice, join them: 3 apples plus 2 apples is 5 apples. Do this to one side at a time. It only changes how the side looks, not what it is worth.
After that, you usually have something like 5x + 6 = 31. You already know that one. Take away the 6, then split by 5. Last, put your answer back into the first equation to make sure it works.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Everyone gets one
You are planning a hangout for 3 friends and you promise each one a movie ticket and a snack. You did not promise 3 tickets and one snack. Every friend gets both, so you owe 3 tickets and 3 snacks.
How it connects: A number multiplying a group multiplies every term inside. If a ticket costs t dollars and a snack costs s dollars, the total for 3 friends is 3(t + s) = 3t + 3s. The sharing only happens when the number multiplies the group. In 3 + (t + s) the 3 is added, so nothing is shared.
The coupon removes the whole combo
Your food order comes to 20 dollars. The cashier applies a coupon that takes off a whole combo, a burger and fries. The coupon removes the burger and the fries, not just the burger.
How it connects: With b and f as the prices in dollars, you pay 20 − (b + f) = 20 − b − f. A minus sign in front of parentheses affects every term inside, which is why 25 − (2x + 3) becomes 25 − 2x − 3. A plus sign in front leaves every sign alone.
Inventory stacking
In many games, identical items stack into one slot: 4 potions and 3 potions become one stack of 7 potions. A sword and a potion never merge into the same stack.
How it connects: Like terms stack: 4x + 3x = 7x. A plain number is a different kind of item, so 7x + 5 cannot be combined further. The kind of item is the variable part, and only the count changes.
Counting in a unit that fits every note
In 4/4 time, a quarter note lasts 1 beat, a half note lasts 2 beats, and an eighth note lasts half a beat. A drummer who wants whole numbers picks a counting unit that fits evenly into every note. Here that is the eighth: it fits into a quarter note 2 times, a half note 4 times, and an eighth note 1 time. So the quarter note is 2 counts, the half note is 4 counts, and the eighth note is 1 count.
How it connects: Multiplying by the common denominator is like picking a counting unit that fits evenly into every piece. For halves and thirds, that unit is a sixth, not a third: a sixth fits into a half 3 times and into a third 2 times. So multiplying x/2 + x/3 = 10 by 6 gives whole-number terms: 3x + 2x = 60. The limit: the unit is not just the smallest piece you see. It must fit evenly into every piece, which is why you multiply by a common denominator (6) and not just the largest denominator (3). And you must recount the whole piece, so every term on both sides gets multiplied, not just one note.
Measure twice, cut once
Carpenters say this because a cut cannot be undone, so they double-check the measurement before they commit.
How it connects: Substituting your answer is the second measurement. Measure against the original plan, the original equation, and not against your own notes, because a slip in the tidied version would not show up there.
Take it one step at a time
This idiom means to handle a big task in small pieces instead of all at once.
How it connects: A long equation is a short list of small moves: distribute, combine, undo, check. Writing one move per line makes a slip easy to find. Once you are confident, small things like combining like terms can share a line.
Knock, knock, parentheses
Why did the 3 knock on the parentheses? It had something to share with everyone inside.
How it connects: The 3 multiplies every term in the parentheses, not just the first one: 3(x + 2) = 3x + 6, not 3x + 2.
Memory help
Dancing Cats Undo Chaos
Run the plan in order: Distribute, Combine like terms, Undo (add or subtract first, then multiply or divide, for equations shaped like ax + b = c), Check. Say it as “Dancing Cats Undo Chaos.” Skip a letter that does not apply: no parentheses means no Distribute, and nothing to join means no Combine.
The tidy-then-undo rhyme
Spread it out, then join the like; next, undo with all your might. Plus or minus goes first, you know; then times or divide, and check it. Go! (The plus-or-minus-first order is for equations shaped like ax + b = c.)
The two cleanup crews
Picture two crews. The Tidy Crew works inside one side of the room at a time. They spread things out (distribute) and sort them (combine like terms), but they never change what the side is worth. The Balance Crew only moves when both sides get the same treatment: add, subtract, multiply or divide, on both sides. Tidy first so the Balance Crew has something clean to work with.
Draw the arcs
Before you distribute, draw an arc from the number outside the parentheses to each term inside. Count the arcs: that is how many products you must write. Keep each sign attached to its term as you draw.
A minus flips everyone inside
A minus sign in front of parentheses flips every sign inside: −(x + 3) = −x − 3, not −x + 3. Say it out loud: “the minus reaches everyone in the group.”
Color-code the like terms
Circle every x-term in one color and box every plain number in another, keeping each sign with its term. Terms marked the same way are like terms. Join the terms that match, and nothing else.
Study notes
Key ideas
- Multi-step equations take more than two moves. Tidy each side first (distribute, then combine like terms), then undo, then check.
- Tidying rewrites one side with the same value, so it is not a balance move. The undo moves must be done to both sides.
- For equations shaped like ax + b = c, undo add or subtract first, then multiply or divide.
- A number or a minus sign in front of parentheses multiplies every term inside, sign included.
- To clear fractions, multiply every term on both sides by a common denominator. To undo a fraction that multiplies a group, multiply both sides by its reciprocal.
- Check in the original equation.
Steps
- If there are fractions, you can clear them first: multiply every term on both sides by a common denominator.
- Distribute any number (with its sign) over every term inside parentheses.
- Combine like terms on each side.
- Undo the add or subtract on both sides.
- Undo the multiply or divide on both sides (never by zero).
- Check by substituting into the original equation.
Formulas
- Distributive property: a(b + c) = ab + ac and a(b − c) = ab − ac
- Minus before parentheses: −(b + c) = −b − c and −(b − c) = −b + c
- Reciprocal: (p/q) · (q/p) = 1 (when p ≠ 0 and q ≠ 0)
Watch out
- Distribute to every term: 3(x + 2) = 3x + 6, not 3x + 2.
- A negative times a negative is positive: −2(x − 5) = −2x + 10.
- A minus sign before parentheses flips every sign inside: 25 − (2x + 3) = 25 − 2x − 3.
- Only like terms combine: 3x + 5 cannot be added into 8x.
- When you clear fractions, multiply every term, including the whole numbers.
- Check in the original equation, not the tidied one.
Why it works
- Why can you tidy one side without touching the other? Distributing and combining only rewrite an expression as an equal expression. Its value does not change, so the balance is untouched.
- Why does distributing work? 3(x + 2) means 3 copies of (x + 2), which hold 3 copies of x and 3 copies of 2.
- Why check in the original equation? A slip made while tidying carries into the tidied equation. Only the original equation tells the truth about the problem.
Common mistakes
Teach it back
Explain to a friend (or a pet!) how to solve 4(x − 1) + 2x = 38 and why you tidy before you undo.
Explaining it out loud shows what you really know. Did you…
- I distributed the 4 to both terms (4x − 4) and combined like terms (6x − 4 = 38).
- I undid the −4 and the ×6 on both sides to get x = 7.
- I checked in the original equation: 4(7 − 1) + 2(7) = 24 + 14 = 38.
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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