A17: Solving Linear Equations
Solve linear equations in one unknown; include unknown on both sides, brackets, fractions
📋 Key Concepts
Linear equation: An equation where the highest power of the variable is 1.
Goal: Find the value of the unknown that makes the equation true. Use inverse operations to isolate the variable.
Golden rule: Do the same to both sides!
📝 One-Step Equations
Example 1
Solve: x + 7 = 12
Solution:
Subtract 7 from both sides: x = 12 - 7
x = 5
Example 2
Solve: 3x = 15
Solution:
Divide both sides by 3: x = 15 ÷ 3
x = 5
Example 3
Solve: x⁄4 = 8
Solution:
Multiply both sides by 4: x = 8 × 4
x = 32
📝 Two-Step Equations
Order: First deal with addition/subtraction, then multiplication/division.
Example 4
Solve: 2x + 5 = 13
Solution:
Step 1: Subtract 5: 2x = 8
Step 2: Divide by 2: x = 4
Example 5
Solve: 4x - 3 = 17
Solution:
Step 1: Add 3: 4x = 20
Step 2: Divide by 4: x = 5
Example 6
Solve: x⁄3 + 2 = 7
Solution:
Step 1: Subtract 2: x⁄3 = 5
Step 2: Multiply by 3: x = 15
📝 Unknown on Both Sides
Method: Collect all x terms on one side and all numbers on the other.
Example 7
Solve: 5x + 2 = 3x + 10
Solution:
Step 1: Subtract 3x: 2x + 2 = 10
Step 2: Subtract 2: 2x = 8
Step 3: Divide by 2: x = 4
Example 8
Solve: 7x - 3 = 4x + 9
Solution:
Subtract 4x: 3x - 3 = 9
Add 3: 3x = 12
Divide by 3: x = 4
Example 9
Solve: 6x + 1 = 2x + 13
Solution:
Subtract 2x: 4x + 1 = 13
Subtract 1: 4x = 12
Divide by 4: x = 3
📝 Equations with Brackets
Method: Expand the brackets first, then solve.
Example 10
Solve: 3(x + 2) = 21
Solution:
Expand: 3x + 6 = 21
Subtract 6: 3x = 15
Divide by 3: x = 5
Example 11
Solve: 2(x - 3) = x + 5
Solution:
Expand: 2x - 6 = x + 5
Subtract x: x - 6 = 5
Add 6: x = 11
Example 12
Solve: 4(2x + 1) = 3(x + 6)
Solution:
Expand: 8x + 4 = 3x + 18
Subtract 3x: 5x + 4 = 18
Subtract 4: 5x = 14
x = 14⁄5 = 2.8
📝 Equations with Fractions
Method: Multiply every term by the common denominator to remove fractions.
Example 13
Solve: x⁄2 + 3 = 7
Solution:
Multiply all terms by 2: x + 6 = 14
Subtract 6: x = 8
Example 14
Solve: x⁄3 + x⁄4 = 7
Solution:
Multiply all terms by 12: 4x + 3x = 84
7x = 84
x = 12
❓ Practice Questions
Q1: Solve: x + 9 = 15
Q2: Solve: 5x = 35
Q3: Solve: 3x - 7 = 11
Q4: Solve: 4x + 3 = 2x + 13
Q5: Solve: 2(x + 4) = 18
Q6: Solve: x⁄5 - 2 = 4
✅ Answers
- x = 6
- x = 7
- x = 6
- x = 5
- x = 5
- x = 30
🎯 Exam Tips
- Do the same to both sides
- Collect x terms on one side, numbers on the other
- Expand brackets before solving
- Multiply by denominator to remove fractions
- Always check your answer by substituting back
🧠 Problem-Solving Strategies
Problem-Solving
For multi-step equations: expand brackets first, then collect like terms, then isolate the variable. For fractions, multiply all terms by the common denominator. Always check your answer by substituting back into the original equation.
Multi-Step Problem
Solve: 3(2x - 1) - 2(x + 4) = 5x + 3
Solution:
Step 1: Expand: 6x - 3 - 2x - 8 = 5x + 3
Step 2: Simplify LHS: 4x - 11 = 5x + 3
Step 3: Collect x terms: -11 - 3 = 5x - 4x → -14 = x
Check: 3(2(-14)-1) - 2((-14)+4) = 3(-29) - 2(-10) = -87 + 20 = -67. RHS: 5(-14) + 3 = -67 ✓
⚠️ Common Errors
Watch Out!
1. Wrong: Expanding -2(x + 4) as -2x + 4 Correct: -2(x + 4) = -2x - 8 (multiply BOTH terms by -2)
2. Wrong: Solving x⁄3 + x⁄4 = 7 by adding numerators: 2x⁄7 = 7 Correct: Multiply all terms by 12 (LCM of 3, 4): 4x + 3x = 84 → 7x = 84 → x = 12
3. Wrong: 4x + 3 = 2x + 9 → 4x + 2x = 9 + 3 Correct: 4x - 2x = 9 - 3 (change signs when moving across the equals sign)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A rectangle has length (2x + 1) cm and width (x + 3) cm. The perimeter equals the area. (a) Write an equation. (b) Solve to find x. (c) Find the dimensions of the rectangle.
(a) Perimeter = 2(2x + 1 + x + 3) = 2(3x + 4) = 6x + 8. Area = (2x + 1)(x + 3) = 2x² + 7x + 3
Equation: 2x² + 7x + 3 = 6x + 8 → 2x² + x - 5 = 0
(b) Using the quadratic formula: x = -1 ± √(1+40)⁄4 = -1 ± √41⁄4. Since x > 0: x = -1 + 6.40⁄4 ≈ 1.35
(c) Length ≈ 2(1.35) + 1 = 3.70 cm, Width ≈ 1.35 + 3 = 4.35 cm
Mark scheme: (a) 2 marks for both expressions and equation. (b) 2 marks. (c) 2 marks for dimensions.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A plumber charges £45 call-out plus £30 per hour. A second plumber charges £20 call-out plus £40 per hour.
(a) Write a formula for each plumber's charge C for h hours.
(b) For how many hours are both plumbers the same cost?
(c) A job takes 3 hours. Which plumber is cheaper and by how much?
Answers: (a) Plumber 1: C = 45 + 30h. Plumber 2: C = 20 + 40h. (b) 45 + 30h = 20 + 40h → 25 = 10h → h = 2.5 hours. (c) P1: £135, P2: £140. Plumber 1 is £5 cheaper.
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