➕ Algebra 1 · Equations
Solving One-Step Equations
Solve one-step equations like x + 7 = 12, x/6 = 5 and −3x = 21 by doing the opposite operation to both sides, with balance pictures, 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.
- 6.EE.B.5: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
- 6.EE.B.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
- A-REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution.
- A-REI.B.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
By the end you can…
- Explain why an equation is a balance, and why you do the same thing to both sides.
- Solve the four one-step forms, x + a = b, x − a = b, ax = b and x/a = b, including ones with negatives, fractions and decimals.
- Check an answer by substituting it back into the original equation.
- Write and solve a one-step equation for a word problem.
Lesson (school style)
An equation is a balance: both sides have the same value. A one-step equation needs just one move to get the variable alone. The move is the opposite of what was done to the variable, and you do it to both sides so the balance holds. The two-step lesson builds on this by chaining two such moves.
Equations and solutions
An equation says two expressions are equal, like x + 7 = 12. A solution is a value of the variable that makes the equation true. Here x = 5 works, because 5 + 7 = 12. No other number does.
Solving means finding the solution. The goal is to get the variable alone on one side, like x = 5.
The balance rule
Picture a scale that is perfectly level. If you take 7 off one pan, you must take 7 off the other pan too, or the scale tips.
Adding or subtracting the same number on both sides, or multiplying or dividing both sides by the same number (not zero), never changes the solution.
Why not zero? Multiplying both sides by 0 turns them into 0 = 0, which is true whatever x is, so it tells you nothing about the solution. Dividing by 0 is not possible at all.
Undo with the opposite
Look at what is stuck to the variable, then do the inverse operation to both sides. Addition and subtraction are inverses. Multiplication and division are inverses.
- x + 7 = 12: 7 is added, so subtract 7 from both sides.
- x − 9 = 4: 9 is subtracted, so add 9 to both sides.
- 4x = 28: x is multiplied by 4, so divide both sides by 4.
- x/6 = 5: x is divided by 6, so multiply both sides by 6.
In general: x + a = b gives x = b − a, and x − a = b gives x = b + a. When a ≠ 0, ax = b gives x = b ÷ a, and x/a = b gives x = a × b.
Negatives, fractions and decimals
The same idea works with any kind of number.
- Negative answers. The solution can be negative: x + 10 = 4 gives x = 4 − 10 = −6.
- Negative coefficients. Divide like usual: −5x = 40 gives x = 40 ÷ (−5) = −8. A lone minus sign counts as −1, so −x = 12 means −1 × x = 12, and x = −12.
- Fractions. To undo multiplying by 7/5, multiply by its reciprocal, 5/7, because (7/5)(5/7) = 1. So (7/5)x = 14 gives x = 14 × 5/7 = 10.
- Decimals. For 0.2x = 3, divide both sides by 0.2. Or notice that 0.2 = 1/5 and multiply both sides by 5. Either way, x = 15.
The variable on the right
The variable does not have to be on the left. In 30 = x + 18 it is on the right. The equal sign works in both directions, so 30 = x + 18 says the same thing as x + 18 = 30.
You can solve it where it is (subtract 18 from both sides to get 12 = x) or flip it first. Either way, x = 12.
Always check
Substitute your answer for the variable in the original equation. If both sides match, the answer is right. If they do not, hunt for the slip. It takes ten seconds.
For the equations in this lesson, an answer that passes the check is the only solution.
Word problems
To turn a story into an equation, name the unknown with a letter and listen for the operation in the words.
- “Gave away 8” or “8 less than” usually means subtracting.
- “Got 5 more” or “5 more than” usually means adding.
- “Times” or “twice” often means multiplying, and so does a total built from equal amounts (“3 dollars each for 5 tickets”). “Split equally” or “each person’s share” often means dividing. “Per” can go either way, so check what is being found.
These words are clues, not rules, so read the whole story before you decide. Watch “less than” in particular. If a ticket costs 4 dollars less than a meal, and t and m are the ticket and meal prices in dollars, then t = m − 4. The meal is the bigger price, so it is the one you start from.
Key words
- Equation
- A statement that two expressions are equal, like x + 7 = 12.
- Solution
- A value of the variable that makes the equation true.
- Inverse operation
- The operation that undoes another one. Addition and subtraction are inverses; so are multiplication and division.
- Isolate
- To get the variable alone on one side of the equation.
- Coefficient
- The number multiplied by a variable. In 4x the coefficient is 4.
- Reciprocal
- The number you multiply a given number by to get 1. The reciprocal of 3/4 is 4/3, because (3/4)(4/3) = 1. Every number except 0 has a reciprocal.
Worked examples
Example 1. Solve x + 7 = 12.
- The variable has 7 added to it. Undo that by subtracting 7 from both sides: x + 7 − 7 = 12 − 7.
- That leaves x = 5.
- Check: 5 + 7 = 12. Both sides match.
Answer: Answer: x = 5
Example 2. Solve x − 9 = 4.
- The variable has 9 taken away. Undo that by adding 9 to both sides: x − 9 + 9 = 4 + 9.
- That leaves x = 13.
- Check: 13 − 9 = 4. Both sides match.
Answer: Answer: x = 13
Example 3. Solve x/6 = 5.
- The variable is divided by 6. Undo that by multiplying both sides by 6: (x/6) × 6 = 5 × 6.
- That leaves x = 30.
- Check: 30/6 = 5. Both sides match.
Answer: Answer: x = 30
Example 4. Solve 4x = 28.
- 4x means 4 times x. Undo that by dividing both sides by 4: 4x ÷ 4 = 28 ÷ 4.
- That leaves x = 7.
- Check: 4 × 7 = 28. Both sides match.
Answer: Answer: x = 7
Example 5. Solve x + 8 = 3.
- Subtract 8 from both sides: x = 3 − 8.
- 3 − 8 = −5, so x = −5. A negative answer is perfectly fine.
- Check: −5 + 8 = 3. Both sides match.
Answer: Answer: x = −5
Example 6. Solve −3x = 21.
- −3x means −3 times x. Undo that by dividing both sides by −3: x = 21 ÷ (−3).
- A positive divided by a negative is negative, so x = −7.
- Check: −3 × (−7) = 21. Both sides match.
Answer: Answer: x = −7
Example 7. Solve (3/4)x = 9.
- The variable is multiplied by 3/4. Multiply both sides by the reciprocal, 4/3, because (4/3)(3/4) = 1.
- The left side becomes x. The right side becomes 9 × 4/3 = 36/3 = 12.
- So x = 12.
- Check: (3/4) × 12 = 36/4 = 9. Both sides match.
Answer: Answer: x = 12
Example 8. Solve 21 = x + 15.
- The variable is on the right, and that is fine. Subtract 15 from both sides: 21 − 15 = x + 15 − 15.
- That leaves 6 = x, which says the same thing as x = 6.
- Check: 6 + 15 = 21. Both sides match.
Answer: Answer: x = 6
Example 9. Mia gave away 8 stickers and has 15 left. How many stickers did she start with?
- Let s be the number she started with. Giving away 8 is subtracting 8, so s − 8 = 15.
- Undo the subtraction by adding 8 to both sides: s = 15 + 8 = 23.
- Check: 23 − 8 = 15. Mia started with 23 stickers.
Answer: Answer: 23 stickers
In plain English
An equation is a number puzzle with a mystery number. For example: “I am thinking of a number. If you add 6, you get 20. What is my number?”
To find it, undo what was done. They added 6, so take 6 away: 20 − 6 = 14. The mystery number is 14.
Every one-step equation works this way. Find the one thing that was done to the mystery number, then do the opposite. Adding is undone by subtracting. Multiplying is undone by dividing.
One rule keeps it fair: whatever you do to one side of the equal sign, you do to the other side too.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Splitting the bill
Four friends split a pizza bill evenly, and each one pays 11 dollars. To find the total, you do the opposite of splitting: you multiply. The total bill is 4 × 11 = 44 dollars.
How it connects: Splitting evenly is dividing by 4: b/4 = 11, where b is the bill in dollars. Undo it by multiplying both sides by 4: b = 44. Check: 44 ÷ 4 = 11. The limit: this works because everyone pays the same share.
Cinderella’s midnight
In Disney’s animated movie Cinderella (1950), based on the fairy tale, the Fairy Godmother’s magic turns a pumpkin into a coach and mice into horses. The spell lasts only until midnight, and when the clock strikes twelve, the coach turns back into a pumpkin and the horses turn back into mice.
How it connects: Adding or subtracting any number, and multiplying or dividing by a nonzero number, each have an undo that sends you back. Adding 5 is undone by subtracting 5, and multiplying by 3 is undone by dividing by 3. (Multiplying by 0 has no undo, which is why you never multiply both sides by 0.) The limit: in an equation you pick the undo yourself, and you must do it to both sides, not just one.
The proof of the pudding is in the eating
This old proverb means you can only judge something by actually trying it. Being told the pudding is good is not enough.
How it connects: The proof of a solution is in the substituting. Suppose you think the solution of x − 6 = 11 is x = 17. Try it: 17 − 6 = 11, so it works. If the sides do not match, the answer is wrong. For the one-step equations in this lesson, an answer that passes the check is the only one.
Level the playing field
To “level the playing field” means to make conditions fair so that nobody has an advantage.
How it connects: An equation starts out level: both sides are equal. A move that treats both sides the same keeps it level, and a move on only one side tips it. That is why you add or subtract the same number on both sides, or multiply or divide both sides by the same number (not zero).
Replay review
Your team’s score on the scoreboard reads 64 right after one of your players makes a 3-point shot. Then the referees watch the replay and cancel the basket. The score drops by exactly 3, to 61.
How it connects: If x was the score before the shot, then x + 3 = 64. Canceling the shot is the inverse move: subtract 3, so x = 64 − 3 = 61. In an equation you do the same to both sides: x + 3 − 3 = 64 − 3.
Quest points
In a game, every quest you finish gives the same number of experience points (XP). Say each quest gives 30 XP, and quests have earned you 120 XP in total. How many quests did you finish?
How it connects: 30q = 120. The quest count is multiplied by 30, so undo it by dividing both sides by 30: q = 4. Check: 30 × 4 = 120.
Sugar by the batch
A cookie recipe uses 2/3 cup of sugar for each batch. When you finish baking, you realize you used 6 cups of sugar in all. How many batches did you make?
How it connects: (2/3)b = 6. Multiply both sides by the reciprocal, 3/2: b = 6 × 3/2 = 9. Check: (2/3)(9) = 6. Nine batches of 2/3 cup fill up 6 cups.
Alone at last
Why did x look so happy at the end of the problem? It finally had one whole side of the equal sign to itself.
How it connects: Solving means isolating the variable: getting x alone on one side, like x = 2. Every step you take moves toward that goal.
Memory help
SOS: spot, opposite, same
Spot the operation stuck to the variable. Do the Opposite operation. Do the Same thing to both sides. Then plug your answer back in. SOS, then check.
The opposites rhyme
Plus and minus undo each other. Times and divide undo each other. Both sides get the move, to keep it fair. Then plug it back in and check it there.
See the scale
Picture a balance scale with x + 7 on the left pan and 12 on the right pan. Take 7 off the left pan and 7 off the right pan, and x is left alone, balanced against 5. If you only took from one pan, the picture tips.
The four-box chart
Draw four boxes. In each one, write a form and its undo move: x + a = b, subtract a. x − a = b, add a. ax = b, divide by a. x/a = b, multiply by a. Cover the undo moves and quiz yourself.
Do the opposite, not the same
In y − 12 = 5, the 12 is already being subtracted, so you add 12. In y/3 = 4, the y is already being divided, so you multiply by 3. Repeating the operation you see only makes the equation messier.
A lone minus sign is −1
In −x = 12 there is a hidden −1 in front of the x. Multiply (or divide) both sides by −1 and you get x = −12. Ignoring the minus sign gives the wrong answer, 12.
Study notes
Key ideas
- An equation is a balance. A solution is a value of the variable that makes it true.
- To solve a one-step equation, do the inverse operation to both sides: addition and subtraction undo each other, and so do multiplication and division.
- Multiply or divide both sides only by numbers that are not zero.
- Always check by substituting the answer into the original equation.
Steps
- Spot the one operation attached to the variable (add, subtract, multiply or divide).
- Do the opposite operation to both sides.
- Make sure the variable is alone on one side. If it ended up on the right, flip the equation.
- Check by substituting into the original equation.
Formulas
- Add form: x + a = b gives x = b − a
- Subtract form: x − a = b gives x = b + a
- Multiply form: ax = b gives x = b ÷ a (when a ≠ 0)
- Divide form: x/a = b gives x = a × b (when a ≠ 0)
- Fraction coefficient: (p/q)x = b gives x = b × (q/p) (when p ≠ 0 and q ≠ 0)
Watch out
- Do the move to both sides, not just one.
- Use the opposite operation. In x − 9 = 4 you add 9; you do not subtract again.
- A lone minus sign is −1: in −x = 12 the answer is x = −12, not 12.
- Mind the signs when you divide: a negative divided by a negative is positive, and a positive divided by a negative is negative.
- Check in the original equation, not in a step you have already changed.
Why it works
- Why do the same thing to both sides? Equal things stay equal when you treat them equally.
- Why not multiply by 0? Both sides become 0, and 0 = 0 is true no matter what x is, so you lose track of the solution. Dividing by 0 is not possible.
- Why use the opposite operation? It cancels what is stuck to x: subtracting 7 cancels adding 7, and multiplying by 4/3 cancels multiplying by 3/4.
Common mistakes
Teach it back
Explain to a friend (or your dog!) how to solve x + 24 = 60, why you do the same thing to both sides, and how to check your answer.
Explaining it out loud shows what you really know. Did you…
- I said to subtract 24 from both sides, because subtracting undoes adding.
- I explained that both sides must get the same move to stay balanced.
- I got x = 36.
- I checked it: 36 + 24 = 60.
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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