Question 5 – Cross Multiplication Method
📚 Understanding Types of Solutions
For a pair of linear equations:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)
| Condition | Type of Solution | Lines |
|---|---|---|
| \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) | Unique solution | Intersecting |
| \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) | No solution | Parallel |
| \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) | Infinitely many solutions | Coincident |
📐 Cross Multiplication Method
For equations: \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \)
Solution:
\[ x = \frac{b_1c_2 – b_2c_1}{a_1b_2 – a_2b_1} \] \[ y = \frac{c_1a_2 – c_2a_1}{a_1b_2 – a_2b_1} \]📑 Question Parts
Part (i) – x – 3y – 3 = 0, 3x – 9y – 2 = 0
\( x – 3y – 3 = 0 \)
\( 3x – 9y – 2 = 0 \)
Step 1: Identify coefficients
\( a_1 = 1, b_1 = -3, c_1 = -3 \)
\( a_2 = 3, b_2 = -9, c_2 = -2 \)
Step 2: Compare ratios
\( \frac{a_1}{a_2} = \frac{1}{3} \)
\( \frac{b_1}{b_2} = \frac{-3}{-9} = \frac{1}{3} \)
\( \frac{c_1}{c_2} = \frac{-3}{-2} = \frac{3}{2} \)
Since: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
Visual Graph Representation
📊 Graphical Solution
Graph showing parallel lines (no intersection)
Part (ii) - 2x + y = 5, 3x + 2y = 8
\( 2x + y = 5 \) or \( 2x + y - 5 = 0 \)
\( 3x + 2y = 8 \) or \( 3x + 2y - 8 = 0 \)
Step 1: Identify coefficients
\( a_1 = 2, b_1 = 1, c_1 = -5 \)
\( a_2 = 3, b_2 = 2, c_2 = -8 \)
Step 2: Compare ratios
\( \frac{a_1}{a_2} = \frac{2}{3} \)
\( \frac{b_1}{b_2} = \frac{1}{2} \)
Since: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
The system has a unique solution.
Step 3: Apply Cross Multiplication Method
Cross Multiplication Diagram
b₁ c₁ a₁
1 -5 2
2 -8 3
b₂ c₂ a₂
Step 4: Verification
LHS = \( 2(2) + 1 = 4 + 1 = 5 \) = RHS ✓
Check in equation (2): \( 3x + 2y = 8 \)LHS = \( 3(2) + 2(1) = 6 + 2 = 8 \) = RHS ✓
Visual Graph Representation
📊 Graphical Solution
Graph showing intersection at (2, 1)
Part (iii) - 3x – 5y = 20, 6x – 10y = 40
\( 3x - 5y = 20 \) or \( 3x - 5y - 20 = 0 \)
\( 6x - 10y = 40 \) or \( 6x - 10y - 40 = 0 \)
Step 1: Identify coefficients
\( a_1 = 3, b_1 = -5, c_1 = -20 \)
\( a_2 = 6, b_2 = -10, c_2 = -40 \)
Step 2: Compare ratios
\( \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2} \)
\( \frac{b_1}{b_2} = \frac{-5}{-10} = \frac{1}{2} \)
\( \frac{c_1}{c_2} = \frac{-20}{-40} = \frac{1}{2} \)
Since: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
Notice that equation (2) is just equation (1) multiplied by 2:
\[ 2 \times (3x - 5y - 20) = 6x - 10y - 40 \]Both equations represent the same line.
Visual Graph Representation
📊 Graphical Solution
Graph showing coincident lines (overlap completely)
Part (iv) - x – 3y – 7 = 0, 3x – 3y – 15 = 0
\( x - 3y - 7 = 0 \)
\( 3x - 3y - 15 = 0 \)
Step 1: Identify coefficients
\( a_1 = 1, b_1 = -3, c_1 = -7 \)
\( a_2 = 3, b_2 = -3, c_2 = -15 \)
Step 2: Compare ratios
\( \frac{a_1}{a_2} = \frac{1}{3} \)
\( \frac{b_1}{b_2} = \frac{-3}{-3} = 1 \)
Since: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
The system has a unique solution.
Step 3: Apply Cross Multiplication Method
Step 4: Verification
LHS = \( 4 - 3(-1) - 7 = 4 + 3 - 7 = 0 \) = RHS ✓
Check in equation (2): \( 3x - 3y - 15 = 0 \)LHS = \( 3(4) - 3(-1) - 15 = 12 + 3 - 15 = 0 \) = RHS ✓
Visual Graph Representation
📊 Graphical Solution
Graph showing intersection at (4, -1)
📊 Summary of All Solutions
| Part | Equations | Type | Solution |
|---|---|---|---|
| (i) | \( x - 3y - 3 = 0 \) \( 3x - 9y - 2 = 0 \) | Parallel | No solution |
| (ii) | \( 2x + y = 5 \) \( 3x + 2y = 8 \) | Intersecting | \( x = 2, y = 1 \) |
| (iii) | \( 3x - 5y = 20 \) \( 6x - 10y = 40 \) | Coincident | Infinite solutions |
| (iv) | \( x - 3y - 7 = 0 \) \( 3x - 3y - 15 = 0 \) | Intersecting | \( x = 4, y = -1 \) |
⚠️ Common Mistakes to Avoid
✅ Correct: Always write in standard form before comparing ratios.
✅ Correct: Be very careful with negative signs in the formula.
✅ Correct: Always compare \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).
✅ Correct: Cross multiplication only works when there's a unique solution.
💡 Key Points to Remember
- Standard Form: Always write as \( ax + by + c = 0 \)
- Ratio Test: Compare \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), \( \frac{c_1}{c_2} \) to determine solution type
- Unique Solution: When \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
- No Solution: When \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
- Infinite Solutions: When \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
- Cross Multiplication: Fast method for unique solutions
- Always Verify: Check your answer in both original equations
Farhan Mansuri
M.Sc. Mathematics, B.Ed.
15+ Years Teaching Experience
Passionate about making mathematics easy and enjoyable for students.