An equation is a balance. Whatever you do to one side you must do to the other — and the whole of this topic is that one sentence, applied carefully.
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Undo the operations in the reverse order to the one that built them up.
⚖️ The order of work
Clear any brackets first: 2(x − 3) = 8 becomes 2x − 6 = 8.
Clear any fractions by multiplying every term by the common denominator.
Gather the letters on one side and the numbers on the other.
Divide by the coefficient — and then check by substituting back.
Multiply every term when clearing a fraction. For x/2 + x/3 = 10, multiplying by 6 gives 3x + 2x = 60, not 3x + 2x = 10. Forgetting the right-hand side is the classic slip.
Checking costs five seconds. Solving 5x − 3 = 2x + 9 gives x = 4. Substitute: 20 − 3 = 17 and 8 + 9 = 17. They match, so the answer is safe.
Name the unknown, write one sentence as one equation, solve, then answer the actual question.
📝 The standard set-ups
Consecutive numbers: x and x + 1. Consecutive even or odd numbers: x, x + 2, x + 4.
Ages: write today’s ages first, then add the same number of years to every person.
Ratio 3 : 5: call them 3x and 5x, never x and 5x/3.
Rectangle: if the length is twice the breadth, write l = 2b and put it into the perimeter formula.
Answer the question that was asked. If you solved for the son’s age but the paper wanted the father’s, the working is right and the mark is lost. Underline what is being asked before you start.
The same equations, dressed up.
Digit problems. A two-digit number with tens digit a and units digit b is 10a + b, not ab. Reversing it gives 10b + a. That single line unlocks every "the digits are reversed" question.
Speed, distance and time. Distance = speed × time. If a car covers the same road at 40 km/h and returns at 60 km/h, the average speed for the whole trip is not 50 — it is 48 km/h, because more time is spent at the slower speed.
Work problems. If A finishes a job in 12 days, A does 1/12 of it in one day. Working with B, who takes 6 days, they do 1/12 + 1/6 = 1/4 per day, so together they finish in 4 days.
12 Olympiad questions on equations and word problems. 🍀
