Linear equations in one unknown

1Summary 重點摘要

ax + b = c is the classic S1 equation: clear the adding/subtracting first, then divide by the coefficient, until x stands alone.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is a linear equation?A linear equation in one unknown has exactly one unknown letter and its highest power is 1. 'One unknown' means a single letter such as the x in 2x + 3 = 7; 'first degree' means no x² or higher term. This is the foundation of equation solving, and fractional equations and word problems later are built on it. To recognise one, check two things: only one letter appears, and the letter carries no square or higher power. Both satisfied, the linear method applies.
2. 2. Basic idea: balanceThe basic idea is balance: an equation is a weighing scale whose two sides must stay equal forever. Whatever operation is done to the left side is done to the right — adding, subtracting, multiplying or dividing alike; as long as both sides move together, the equation still holds. The goal is to isolate x step by step until x alone equals a number. Keep the scale picture in mind and every later rule has a reason; acting on one side only tips the scale and ruins the answer.
3. 3. Step 1: Remove constantStep one removes the constant: move the number with no x across the equals sign. In 2x + 3 = 7 the constant is +3, so subtract 3 from both sides: 2x + 3 − 3 = 7 − 3, simplifying to 2x = 4. The convention is constants first, coefficients later — addition and subtraction before multiplication and division. The usual mistakes are subtracting on one side only, or forgetting to flip the sign when moving a term; after each line, evaluate both sides separately to confirm they are still equal.
4. 4. Step 2: Remove coefficientStep two removes the coefficient: once the equation reads 2x = 4, a multiplier of 2 still stands before x. Divide both sides by 2: 2x ÷ 2 = 4 ÷ 2, giving x = 2, and the unknown stands fully isolated. Keep the order — constants before coefficients, matching 'add and subtract before multiply and divide'; reversing it, say dividing 2x + 3 = 7 by 2 first, produces fractions and needless trouble. Division must hit both sides; dividing one side only is the classic slip.
5. 5. CheckChecking substitutes the answer into the original equation: with x = 2, 2x + 3 = 2 × 2 + 3 = 4 + 3 = 7, matching the right side, so the solution is correct. Always check against the original equation, not an intermediate line, so that every earlier step is verified together. The check takes seconds and intercepts almost every arithmetic error; verifying each question as it is finished beats verifying everything at the end, because an error caught early costs least to fix.
6. 6. Two-step: with bracketsFor a two-step equation with brackets, expand first. In 3(x + 1) = 12, multiply every term inside by 3: 3 × x + 3 × 1 gives 3x + 3 = 12; second, remove the constant by subtracting 3 from both sides to get 3x = 9; third, divide both sides by 3 to get x = 3. Expanding is where the second term gets dropped — writing 3x + 1 = 12 is the most frequent error of the chapter; the number before the brackets multiplies every single term inside.
7. 7. Two-step: x on both sidesWhen x appears on both sides, as in 2x + 3 = x + 7, first remove the smaller x term: subtract x from both sides to get x + 3 = 7; then remove the constant by subtracting 3, giving x = 4. Removing the smaller term keeps the remaining coefficient positive and reduces sign errors. Check: left side 2 × 4 + 3 = 11, right side 4 + 7 = 11, equal, so the answer is correct. Such equations add just one move — shifting the x — and otherwise follow the two-step method exactly.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Linear: one unknown, first degree
• Equation is balance, both sides equal
• First constant (add/subtract), then coefficient (multiply/divide)
• With brackets: expand first
• Check: substitute back