Linear equations Notes, Quizzes & Revision

• Understand what a linear equation is.
• Solve simple one-step and two-step linear equations in one variable.
• Use the balance (transposition) method and check answers.
• Solve simple word problems that lead to a linear equation.

• Variable — a letter that represents an unknown number (example: x).
• Coefficient — number multiplied by the variable (in 3x, 3 is the coefficient).
• Constant — a number on its own (example: 5 in x + 5 = 12).
• Linear equation — an equation where the highest power of the variable is 1 (example: ax + b = c).

A linear equation in one variable looks like ax + b = c. Example: 2x + 3 = 11. The goal is to find the value of x that makes the statement true.

Think of an equation as a scale that must stay balanced. Whatever you do to one side, do the same to the other.

1. One-step equation: x + 5 = 12.
   Subtract 5 from both sides: x = 12 − 5 = 7. Check: 7 + 5 = 12 ✓
2. Two-step equation: 3x − 4 = 11.
   Add 4 to both sides: 3x = 15. Divide both sides by 3: x = 15 ÷ 3 = 5. Check: 3×5 − 4 = 15 − 4 = 11 ✓
3. Variable on both sides: 2x + 3 = x + 8.
   Subtract x from both sides: x + 3 = 8. Then subtract 3: x = 5. Check: 2×5 + 3 = 10 + 3 = 13 and x + 8 = 5 + 8 = 13 ✓
4. Fraction: (1/2)x + 3 = 7.
   Subtract 3: (1/2)x = 4. Multiply both sides by 2: x = 8. Check: (1/2)×8 + 3 = 4 + 3 = 7 ✓

• Always perform the same operation on both sides.
• Do addition/subtraction first (to move constants), then multiply/divide to isolate the variable.
• Check your answer by substituting back into the original equation.
• Keep equations tidy: collect like terms before dividing.

1. x + 6 = 14
2. 5x = 35
3. 4x − 7 = 13
4. 2x + 9 = x + 17
5. (1/3)x + 5 = 9
6. Amina had some coins. She found 8 more coins and now has 21. How many did she have? (Let x = original coins)
7. 7x − 2 = 20
8. 3(x + 2) = 18

1. x = 8
2. x = 7
3. 4x − 7 = 13 → 4x = 20 → x = 5
4. 2x + 9 = x + 17 → x = 8
5. (1/3)x + 5 = 9 → (1/3)x = 4 → x = 12
6. x + 8 = 21 → x = 13 coins
7. 7x − 2 = 20 → 7x = 22 → x = 22/7 (not an integer)
8. 3(x + 2) = 18 → x + 2 = 6 → x = 4

Linear equations in one variable give single number answers (like x = 5). If you learn linear equations in two variables (for example y = 2x + 1) you will see they make straight lines on a graph — this is the next step in algebra.