Question 4 – Check consistency and obtain graphical solutions
📑 Question Parts
📚 Important Concept
This question combines two skills:
- Algebraic Check: Use ratio method to determine consistency
- Graphical Solution: If consistent, plot the lines and find the solution
Remember: The graphical solution is the point where the two lines intersect (if they do).
Part (i)
\( x + y = 5 \)
\( 2x + 2y = 10 \)
Step 1: Check Consistency Using Ratios
\( x + y – 5 = 0 \)
\( 2x + 2y – 10 = 0 \)
\( a_1 = 1 \), \( b_1 = 1 \), \( c_1 = -5 \)
\( a_2 = 2 \), \( b_2 = 2 \), \( c_2 = -10 \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{2} \)
The lines are coincident (same line).
The system is CONSISTENT with infinitely many solutions.
Step 2: Graphical Solution
| x | 0 | 5 | 3 |
|---|---|---|---|
| y | 5 | 0 | 2 |
| Points | (0, 5) | (5, 0) | (3, 2) |
Simplify: \( x + y = 5 \) (dividing by 2)
This gives the same points as the first equation!
| x | 0 | 5 | 3 |
|---|---|---|---|
| y | 5 | 0 | 2 |
| Points | (0, 5) | (5, 0) | (3, 2) |
Solution: Any point on the line \( x + y = 5 \) is a solution. For example: (0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0), etc.
Part (ii)
\( 3x – 3y = 16 \)
Step 1: Check Consistency Using Ratios
\( x – y – 8 = 0 \)
\( 3x – 3y – 16 = 0 \)
\( a_1 = 1 \), \( b_1 = -1 \), \( c_1 = -8 \)
\( a_2 = 3 \), \( b_2 = -3 \), \( c_2 = -16 \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{1}{3} \) but \( \frac{c_1}{c_2} = \frac{1}{2} \)
\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
The lines are parallel.
The system is INCONSISTENT with no solution.
Step 2: Verification
From first equation: \( x – y = 8 \)
If we multiply by 3: \( 3x – 3y = 24 \)
But the second equation is: \( 3x – 3y = 16 \)
Since \( 24 \neq 16 \), the equations are contradictory. The lines are parallel and never meet.
Part (iii)
\( 4x – 2y – 4 = 0 \)
Step 1: Check Consistency Using Ratios
\( a_1 = 2 \), \( b_1 = 1 \), \( c_1 = -6 \)
\( a_2 = 4 \), \( b_2 = -2 \), \( c_2 = -4 \)
Since \( \frac{a_1}{a_2} = \frac{1}{2} \) and \( \frac{b_1}{b_2} = \frac{-1}{2} \)
\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
The lines intersect at one point.
The system is CONSISTENT with a unique solution.
Step 2: Graphical Solution
| x | 0 | 3 | 2 |
|---|---|---|---|
| y | 6 | 0 | 2 |
| Points | (0, 6) | (3, 0) | (2, 2) |
| x | 0 | 1 | 2 |
|---|---|---|---|
| y | -2 | 0 | 2 |
| Points | (0, -2) | (1, 0) | (2, 2) |
Observing the tables, both lines pass through the point (2, 2).
This is the point of intersection.
Check in first equation: \( 2(2) + 2 – 6 = 4 + 2 – 6 = 0 \) ✓
Check in second equation: \( 4(2) – 2(2) – 4 = 8 – 4 – 4 = 0 \) ✓
Solution: x = 2, y = 2
Part (iv)
\( 4x – 4y – 5 = 0 \)
Step 1: Check Consistency Using Ratios
\( a_1 = 2 \), \( b_1 = -2 \), \( c_1 = -2 \)
\( a_2 = 4 \), \( b_2 = -4 \), \( c_2 = -5 \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{1}{2} \) but \( \frac{c_1}{c_2} = \frac{2}{5} \)
\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
The lines are parallel.
The system is INCONSISTENT with no solution.
Step 2: Verification
Simplify first equation by dividing by 2: \( x – y – 1 = 0 \) or \( x – y = 1 \)
Simplify second equation by dividing by 4: \( x – y – \frac{5}{4} = 0 \) or \( x – y = \frac{5}{4} \)
Since \( 1 \neq \frac{5}{4} \), the equations are contradictory. The lines are parallel.
📊 Summary of All Four Parts
| Part | Equations | Condition | Nature | Consistency | Solution |
|---|---|---|---|---|---|
| (i) | x + y = 5 2x + 2y = 10 | \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) | Coincident | Consistent | Infinitely many |
| (ii) | x – y = 8 3x – 3y = 16 | \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) | Parallel | Inconsistent | No solution |
| (iii) | 2x + y – 6 = 0 4x – 2y – 4 = 0 | \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) | Intersecting | Consistent | x = 2, y = 2 |
| (iv) | 2x – 2y – 2 = 0 4x – 4y – 5 = 0 | \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) | Parallel | Inconsistent | No solution |
⚠️ Common Mistakes to Avoid
✅ Correct: Always use the ratio method first to determine if a graphical solution exists.
✅ Correct: Choose a suitable scale (like 1 cm = 1 unit) and plot at least 3 points per line for accuracy.
✅ Correct: Coincident lines overlap completely (infinitely many solutions), while intersecting lines meet at exactly one point (unique solution).
✅ Correct: Always substitute your graphical solution back into both original equations to confirm.
💡 Key Points to Remember
- Two-Step Process: First check consistency algebraically, then solve graphically if consistent
- Intersecting Lines: One unique solution point
- Coincident Lines: Infinitely many solutions (same line)
- Parallel Lines: No solution (inconsistent)
- Graph Accuracy: Use graph paper, ruler, and appropriate scale
- Point Selection: Choose convenient values (including 0) for easy plotting
- Verification: Always verify graphical solutions algebraically
- Table Method: Create a table of at least 3 points for each line
- Intersection: The solution is where the two lines cross
📝 Practice Similar Problems
Farhan Mansuri
M.Sc. Mathematics, B.Ed.
15+ Years Teaching Experience
Passionate about making mathematics easy and enjoyable for students.