⚔️ Algebra Kingdom · Algebra
One-Step Equations
Solve equations that need a single inverse operation, keeping the balance by doing the same thing to both sides.
In short
- An equation is a balance; any operation applied to both sides keeps it balanced and keeps the same solution.
- To free the unknown, undo what was done to it with the inverse operation: add undoes subtract, multiply undoes divide.
- 3x means 3 times x, so it is undone by dividing by 3 — never by subtracting 3.
- Substituting your answer back into the original equation proves it right or catches the mistake.
An equation is a balance
An expression such as x + 7 is a recipe: it tells you what to do with a number. An equation such as x + 7 = 19 is a claim: it says two recipes give the same result.
Picture an old market balance with a pan on each side. The left pan holds an unknown weight x and a 7 gram weight; the right pan holds 19 grams. The pans are level, and that is what the equals sign means.
Solving means finding the value of x that keeps the pans level — and there is exactly one such value here.
Do the same thing to both sides
If the pans are level and you take 7 grams off one side, they tip. If you take 7 grams off both sides, they stay level.
That is the single rule behind all of algebra:
- add the same number to both sides
- subtract the same number from both sides
- multiply both sides by the same number
- divide both sides by the same number (never by zero)
Each move produces a new equation with exactly the same solution as the old one. You keep making moves until the unknown is standing alone.
Choosing the inverse operation
Look at what has been done to the unknown, then do the opposite.
- x + a = b — a is being added, so subtract a: x = b - a
- x - a = b — a is being subtracted, so add a: x = b + a
- ax = b — x is being multiplied by a, so divide by a: x = b / a
- x / a = b — x is being divided by a, so multiply by a: x = ab
Addition and subtraction are a pair; multiplication and division are a pair. Notice that ax means "a times x", so it is never undone by subtracting a — a product is not a sum.
A number line helps for the first two: x + 5 = 3 asks "from where do I land on 3 after moving 5 to the right?" You have to start at -2.
Always check
A solution is a claim you can test, which makes algebra unusually forgiving: substitute your answer into the original equation and see whether both sides really do come out the same.
If x = -12 is offered for x + 17 = 5, put it back: -12 + 17 = 5. True, so the answer is right.
Checking catches sign slips almost every time, and it costs a few seconds. Do it in your head if you can, on paper if the numbers are large.
From a story to an equation
Word problems become one-step equations once you name the unknown.
1. Write "let n be ..." and say exactly what n counts. 2. Follow the story forwards, writing what happens to n. 3. The sentence that gives a total or a result is where the equals sign goes. 4. Solve, then answer in words with the right unit.
"Yuan set out with a pouch of coins, handed over 26 of them, and had 41 left" becomes n - 26 = 41 and so n = 67 coins. Reading it forwards keeps the operation the right way round — the story subtracts, so working backwards adds.
Worked examples
Example 1
Solve for x: x + 17 = 5
- Something is being added to x, so the inverse operation is subtraction.
- Subtract 17 from both sides: x + 17 - 17 = 5 - 17.
- The left side collapses to x, because +17 - 17 is 0.
- x = 5 - 17 = -12.
- Check: -12 + 17 = 5, which is what the equation claimed. Correct.
Example 2
A haul of crystals is split into 7 equal shares, and each share holds 24 crystals. How many crystals were in the haul?
- Let n be the number of crystals in the whole haul.
- Splitting into 7 equal shares is written n/7, and the story says one share is 24: n/7 = 24.
- n is being divided by 7, so multiply both sides by 7: n = 24 x 7.
- n = 168 crystals.
- Check: 168 / 7 = 24, the size of one share. Correct.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5Solve for x: x + 6 = 9
Answer: 3
- x + 6 = 9
- x + 6 - 6 = 9 - 6 (subtract 6 from both sides)
- x = 3
- Check: 3 + 6 = 9, which is exactly what the equation says.
Problem 2
Difficulty 3 of 5Solve for w: w - 4 = 39
Answer: 43
- w - 4 = 39
- w - 4 + 4 = 39 + 4 (add 4 to both sides)
- w = 43
- Check: 43 - 4 = 39. The balance holds.
Problem 3
Difficulty 4 of 5Solve for w: 16w = -48
Answer: -3
- 16w = -48
- 16w / 16 = -48 / 16 (divide both sides by 16)
- w = -3
- Check: 16 x (-3) = -48. Correct.
Common mistakes
- Using the wrong inverse: answering 26 for x + 19 = 7 because the two numbers were added instead of subtracted.
- Changing only one side of the equation, which breaks the balance.
- Answering with the number on the right of the equals sign instead of the value of the unknown.
- Subtracting the coefficient in 5x = 40 and answering 35 instead of dividing to get 8.
What you should be able to do
- Solve x + a = b and x - a = b.
- Solve ax = b and x / a = b.
- Check a solution by substituting it back.
- Write a one-step equation from a word problem.
Where this fits in the curriculum
Common Core
Ontario
- G4.C2.2
Grade 4 — Determine the unknown in equations involving a single operation, and verify the solution.
- G5.C2.3
Grade 5 — Solve equations that involve one operation and whole numbers, and verify the solution.
Grade 6 is already past this skill: its C2.3 asks for equations with multiple terms, so "one step" sits at Grades 4–5 in Ontario rather than at Grade 6.