Reference · module 6
Equations
Everything this module explains, on one page.
Finding the unknown
Solve simple equations by undoing what was done to x.
- An equation says two sides are equal, and x stands for a number we do not know yet. x + 7 = 12 asks: what number plus 7 gives 12? The answer, 5, is the solution.
- Undo each operation with its opposite, doing the same to both sides. x + 7 = 12 becomes x = 12 − 7. 3x = 18 becomes x = 18 ÷ 3. The sides stay balanced, like scales.
- Always check by putting the answer back. If x = 6 solves 3x = 18, then 3 × 6 must be 18 — and it is. A check takes seconds and catches most slips.
The mistake you are about to make
x − 4 = 10 → x = 6
x − 4 = 10 → x = 14
The minus in the equation invites another subtraction. To undo taking away 4, you add 4 back to both sides.
Two-step equations
Solve equations with several operations and x on both sides.
- For 2x + 3 = 11, undo in reverse order: first the + 3, then the × 2. Subtract 3 to get 2x = 8, then divide by 2 to get x = 4.
- When x is on both sides, gather the x's on one side. 5x = 2x + 12 becomes 3x = 12 by taking 2x from both sides, so x = 4.
- Brackets can be opened first: 3(x − 2) = 15 is 3x − 6 = 15, so 3x = 21 and x = 7. Or divide both sides by 3 at once: x − 2 = 5.
The mistake you are about to make
3(x − 2) = 3x − 2
3(x − 2) = 3x − 6
The 3 sits right next to the x, so it seems to belong to the x only. Everything inside the brackets is multiplied, the 2 included.
From words to equations
Turn a problem in words into an equation and solve it.
- Name the unknown first: let x be the price of one ticket. Then translate the sentence piece by piece. Three tickets and a 4-euro fee cost 31: 3x + 4 = 31, so x = 9.
- Consecutive numbers follow one another, so they are x, x + 1 and x + 2. If three consecutive numbers add up to 48, then 3x + 3 = 48 and the numbers are 15, 16 and 17.
- When one amount is described through another, use the same x for both. Ana has 5 more than Bo, and together they have 35: x + (x + 5) = 35, so Bo has 15 and Ana has 20.
The mistake you are about to make
Ana has 5 more: 35 ÷ 2 + 5
x + (x + 5) = 35
Splitting the total in half and then adding the difference feels quick. It gives 22.5 for Ana, and 22.5 plus 17.5 is 40, not 35.