Forming equations from problems

How to turn a worded setup into an equation: one letter for one unknown NUMBER, every other quantity built from it, and one fact that ties the expressions together — then answer the question that was asked.

Algebra · Linear equations

Forming equations from problems

Why it works

Exam problems rarely say "solve this equation" — they hand you a story and expect you to build the equation. The recipe has three moves.

1. One letter, standing for one unknown NUMBER. "Let

x

= Kate's age in years" — not "

x

= Kate". The letter is a number in disguise, so it can be doubled, added, compared. Pick the quantity everything else hangs off (usually the one described last or least).

2. Build every other quantity FROM that letter. "Dan is twice as old as Kate" → Dan is

2x

. "Sam is 4 years younger than Kate" →

x - 4

. Read relationships carefully and test with an easy number: if Kate were 10, Dan is 20 — so Dan

= 2x

, not

2 \times \text{Dan} = x

. Reversing a relationship is the single most common forming error.

3. Find the fact that ties them together — that's the equation. A total ("their ages sum to 44"), a geometric fact (angles in a triangle sum to

180°

, perimeter equals 34), or an equality ("the results are the same"). Then

x + 2x + (x - 4) = 44 \;\Rightarrow\; 4x - 4 = 44 \;\Rightarrow\; x = 12.

Structure captures patterns too. Three consecutive odd numbers:

n

n + 2

n + 4

(odd numbers step by 2). A rectangle's perimeter: two lengths AND two widths,

2(2x + 3) + 2(x - 1)

.

Then answer the QUESTION, not the equation.

x = 12

is Kate — if the question asked for Dan's age, the answer is

24

. After solving, reread the question line and substitute back into the story as a sense check:

12 + 24 + 8 = 44

. ✓