Linear Equations in One Variable
1. What is a Linear Equation?
A linear equation in one variable is an equation where:
- The variable has exponent 1.
- It can be written in the form ax + b = 0, where a is not 0.
Examples of linear equations:
- 2x + 5 = 13
- 3y - 7 = 14
- 4x + 2 = 3x + 8
Non-examples: x^2 + 3 = 7 (x has exponent 2), 2/x = 5 (x in denominator).
2. Solving Equations: Balance Method
An equation is like a balanced weighing scale. Whatever you do to one side, do the same to the other side.
Rule 1: Adding the same number to both sides preserves equality.
Rule 2: Subtracting the same number from both sides preserves equality.
Rule 3: Multiplying both sides by the same (non-zero) number preserves equality.
Rule 4: Dividing both sides by the same (non-zero) number preserves equality.
Worked Example: Solve x + 7 = 15.
Subtract 7 from both sides: x + 7 - 7 = 15 - 7.
x = 8.
Check: 8 + 7 = 15 (correct).
Worked Example: Solve 3x = 21.
Divide both sides by 3: 3x/3 = 21/3.
x = 7.
Check: 3 x 7 = 21 (correct).
3. Transposition Method
Moving a term from one side to the other changes its sign.
- Addition becomes subtraction and vice versa.
- Multiplication becomes division and vice versa.
Worked Example: Solve 5x + 8 = 33.
Transpose 8: 5x = 33 - 8 = 25.
Transpose 5 (multiply -> divide): x = 25/5 = 5.
Check: 5(5) + 8 = 25 + 8 = 33 (correct).
Common Mistake: Forgetting to change the sign when transposing. Example: x + 5 = 12 => x = 12 - 5, not x = 12 + 5.
4. Types of Equations
| Type | Example | Approach |
|---|---|---|
| Simple | 2x + 5 = 13 | Isolate x |
| Variable both sides | 3x + 2 = x + 10 | Bring variables to one side |
| With brackets | 2(x + 3) = 14 | Expand brackets first |
| With fractions | x/3 + 2 = 5 | Multiply by LCM |
Worked Example: Solve 4x - 3 = 2x + 7.
Bring x terms to LHS: 4x - 2x - 3 = 7.
2x - 3 = 7. Transpose -3: 2x = 10.
x = 5.
Check: 4(5) - 3 = 17, 2(5) + 7 = 17 (correct).
Worked Example: Solve 2(x - 3) + 5 = 13.
Expand: 2x - 6 + 5 = 13 => 2x - 1 = 13.
2x = 14 => x = 7.
Check: 2(7 - 3) + 5 = 2(4) + 5 = 8 + 5 = 13 (correct).
5. Word Problems
Step 1: Read the problem carefully.
Step 2: Let the unknown be x.
Step 3: Form an equation from the given information.
Step 4: Solve the equation.
Step 5: Verify the answer with the problem statement.
Exam Focus (4 marks): 'Sum of two numbers is 36. One number is 8 more than the other. Find the numbers.'
Let the smaller number be x. The larger is x + 8.
Equation: x + (x + 8) = 36 => 2x + 8 = 36.
2x = 28 => x = 14.
Numbers: 14 and 22.
Check: 14 + 22 = 36, and 22 - 14 = 8 (correct).
Worked Example: 'The perimeter of a rectangle is 48 cm. The length is 6 cm more than the breadth. Find the dimensions.'
Let breadth = x cm. Length = (x + 6) cm.
Perimeter = 2(length + breadth) = 2(x + 6 + x) = 2(2x + 6) = 4x + 12.
Equation: 4x + 12 = 48 => 4x = 36 => x = 9.
Breadth = 9 cm, Length = 15 cm.
Check: 2(15 + 9) = 2(24) = 48 (correct).
Common Mistake: Forgetting to multiply both length and breadth by 2 in perimeter. Just (length + breadth) gives semi-perimeter, not perimeter.
6. Self-Test
- Solve: (a) 3x - 7 = 14 (b) 6x + 2 = 3x + 11.
- Solve: 4(x - 1) + 3 = 23.
- Solve: x/5 - 2 = 3.
- The sum of three consecutive numbers is 51. Find the numbers.
- A number when multiplied by 5 and then 8 is added gives 48. Find the number.
- The length of a rectangle is 3 times its breadth. If perimeter is 64 cm, find the dimensions.
- The sum of ages of a father and son is 45 years. Father is 25 years older. Find their ages.
7. Answers to Self-Test
- (a) 3x = 21 => x = 7. (b) 6x - 3x = 11 - 2 => 3x = 9 => x = 3.
- 4x - 4 + 3 = 23 => 4x - 1 = 23 => 4x = 24 => x = 6.
- x/5 = 5 => x = 25.
- Let numbers be x, x+1, x+2. 3x + 3 = 51 => 3x = 48 => x = 16. Numbers: 16, 17, 18.
- 5x + 8 = 48 => 5x = 40 => x = 8.
- Let breadth = x. Length = 3x. 2(3x + x) = 8x = 64 => x = 8. Breadth = 8 cm, Length = 24 cm.
- Let son's age = x. Father's age = x + 25. x + x + 25 = 45 => 2x = 20 => x = 10. Son = 10, Father = 35.
