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Foundations · Algebra

Solving simple equations

One-step and two-step equations with the balance idea: undo what was done to x, in the opposite order, on both sides.

First, the rules

Keep the balance, undo in reverse

An equation says two things weigh the same. Whatever you do to one side you do to the other, and you undo the operations that were done to x in the opposite order.

★ Inverse operations
Example 1
+ undoes − · × undoes ÷ · and the other way round.
★ One step
Example 1
x + 7 = 12 → x = 12 − 7 = 5
4x = 28 → x = 28 ÷ 4 = 7
x3 = 6 → x = 18
★ Two steps
Example 2
3x + 5 = 20 → x was multiplied by 3, then 5 was added; undo in reverse: subtract 5 (3x = 15), then divide by 3 (x = 5).
★ x on both sides
Example 3
Move the smaller x term across: 5x − 4 = 2x + 11 → 3x − 4 = 11 → 3x = 15 → x = 5.
Brackets first
Example 3
Expand, then solve: 2(x − 3) = 10 → 2x − 6 = 10 → x = 8 (or divide both sides by 2 first: x − 3 = 5).
★ Always check
Examples 1–3
Substitute back into the original equation. 5 · 5 − 4 = 21 and 2 · 5 + 11 = 21 ✓.

See it

The balance

Boxes hide the unknown x, circles are unit weights, and the pans balance. Press "Next move": loose weights come off both sides, then the right side is shared among the boxes.

Step by step

Worked examples

1

One-step equations

inverse operations

Solve (a) x − 9 = 4, (b) −6x = 42, (c) x5 = −3.

Ask yourself: What was done to x? Do the opposite to both sides.

Show the solution
  1. (a) Add 9: x = 13.
  2. (b) Divide by −6: x = −7.
  3. (c) Multiply by 5: x = −15.

Answer 13 · −7 · −15

2

Two-step equations

reverse order

Solve (a) 4x − 7 = 21 and (b) x2 + 3 = 11.

Ask yourself: The last thing done to x is the first thing you undo.

Show the solution
  1. (a) Add 7: 4x = 28. Divide by 4: x = 7. Check: 28 − 7 = 21 ✓.
  2. (b) Subtract 3: x/2 = 8. Multiply by 2: x = 16. Check: 8 + 3 = 11 ✓.

Answer 7 · 16

3

Both sides, and brackets

both sides

Solve 3(2x + 1) = 4x + 13.

Ask yourself: Expand, collect x on one side and numbers on the other.

Show the solution
  1. Expand: 6x + 3 = 4x + 13.
  2. Subtract 4x: 2x + 3 = 13. Subtract 3: 2x = 10. Divide: x = 5.
  3. Check: 3(11) = 33 and 20 + 13 = 33 ✓.

Answer x = 5

Practice: solve for x

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One-step, two-step, fraction and both-sides equations — type the value of x. Your best score is kept in this browser only.

About this lesson

Solving simple equations with the balance model: one-step equations by undoing an addition or a multiplication, two-step equations by undoing in reverse order, unknowns on both sides, and checking the answer by substitution. A balance you can operate step by step, worked examples and practice questions. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.