Question 6 – Draw pairs of linear equations with specific properties
📑 Question Parts
📚 Understanding the Question
This question asks you to:
- Part (i): Draw two intersecting lines and find the triangle they form with the x-axis
- Part (ii): Draw two parallel lines (lines that never meet)
- Part (iii): Draw two coincident lines (lines that overlap completely)
Key Skill: Understanding different types of line relationships and representing them graphically.
Part (i) – Intersecting Lines Forming a Triangle
Step 1: Find Points for First Line
| x | -1 | 0 | 2 |
|---|---|---|---|
| y = x + 1 | 0 | 1 | 3 |
| Points | (-1, 0) | (0, 1) | (2, 3) |
Note: Point (-1, 0) is where this line crosses the x-axis.
Step 2: Find Points for Second Line
| x | 0 | 4 | 2 |
|---|---|---|---|
| y = (12 – 3x)/2 | 6 | 0 | 3 |
| Points | (0, 6) | (4, 0) | (2, 3) |
Note: Point (4, 0) is where this line crosses the x-axis.
Step 3: Find the Intersection Point
Both lines pass through the point (2, 3).
This is the point where the two lines intersect.
We can verify by solving the equations simultaneously:
From \( x – y + 1 = 0 \): \( x = y – 1 \)
Substitute in \( 3x + 2y – 12 = 0 \):
\[ 3(y – 1) + 2y – 12 = 0 \] \[ 3y – 3 + 2y – 12 = 0 \] \[ 5y = 15 \] \[ y = 3 \]Then \( x = 3 – 1 = 2 \)
Intersection point: (2, 3) ✓
Step 4: Identify the Three Vertices of the Triangle
- Vertex A: Where first line meets x-axis = (-1, 0)
- Vertex B: Where second line meets x-axis = (4, 0)
- Vertex C: Where both lines intersect = (2, 3)
- Plot the line \( x – y + 1 = 0 \) passing through (-1, 0), (0, 1), and (2, 3)
- Plot the line \( 3x + 2y – 12 = 0 \) passing through (0, 6), (4, 0), and (2, 3)
- The two lines intersect at (2, 3)
- The triangle ABC is formed with vertices at A(-1, 0), B(4, 0), and C(2, 3)
- Shade the triangular region ABC
The three vertices of the triangle are:
A(-1, 0), B(4, 0), and C(2, 3)
Part (ii) – Parallel Lines
Understanding Parallel Lines
Two lines are parallel if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]This means they have the same slope but different y-intercepts.
Example: Creating Parallel Lines
Let’s create two parallel lines:
Line 1: \( x + y = 4 \) or \( x + y – 4 = 0 \)
Line 2: \( x + y = 6 \) or \( x + y – 6 = 0 \)
Check if parallel:
\( a_1 = 1, b_1 = 1, c_1 = -4 \)
\( a_2 = 1, b_2 = 1, c_2 = -6 \)
\( \frac{a_1}{a_2} = \frac{1}{1} = 1 \)
\( \frac{b_1}{b_2} = \frac{1}{1} = 1 \)
\( \frac{c_1}{c_2} = \frac{-4}{-6} = \frac{2}{3} \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the lines are parallel ✓
Step 1: Find Points for First Line
| x | 0 | 4 | 2 |
|---|---|---|---|
| y = 4 – x | 4 | 0 | 2 |
| Points | (0, 4) | (4, 0) | (2, 2) |
Step 2: Find Points for Second Line
| x | 0 | 6 | 3 |
|---|---|---|---|
| y = 6 – x | 6 | 0 | 3 |
| Points | (0, 6) | (6, 0) | (3, 3) |
- Plot the line \( x + y = 4 \) passing through (0, 4), (4, 0), and (2, 2)
- Plot the line \( x + y = 6 \) passing through (0, 6), (6, 0), and (3, 3)
- Both lines have the same slope (-1) but different y-intercepts
- The lines run parallel to each other and never intersect
- The perpendicular distance between them remains constant
Example of parallel lines:
\( x + y = 4 \) and \( x + y = 6 \)
These lines have the same slope but never intersect.
💡 Other Examples of Parallel Lines:
- \( 2x + 3y = 6 \) and \( 2x + 3y = 12 \)
- \( y = 2x + 1 \) and \( y = 2x + 5 \)
- \( 3x – 4y = 8 \) and \( 3x – 4y = 16 \)
Part (iii) – Coincident Lines
Understanding Coincident Lines
Two lines are coincident if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]This means they represent the same line (one equation is a multiple of the other).
Example: Creating Coincident Lines
Let’s create two coincident lines:
Line 1: \( 2x + 3y = 6 \) or \( 2x + 3y – 6 = 0 \)
Line 2: \( 4x + 6y = 12 \) or \( 4x + 6y – 12 = 0 \)
Check if coincident:
\( a_1 = 2, b_1 = 3, c_1 = -6 \)
\( a_2 = 4, b_2 = 6, c_2 = -12 \)
\( \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \)
\( \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \)
\( \frac{c_1}{c_2} = \frac{-6}{-12} = \frac{1}{2} \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{2} \), the lines are coincident ✓
Verification: If we multiply the first equation by 2:
\[ 2(2x + 3y – 6) = 0 \] \[ 4x + 6y – 12 = 0 \]This is exactly the second equation! They represent the same line.
Step 1: Find Points for First Line
| x | 0 | 3 | -3 |
|---|---|---|---|
| y = (6 – 2x)/3 | 2 | 0 | 4 |
| Points | (0, 2) | (3, 0) | (-3, 4) |
Step 2: Find Points for Second Line
Simplify: \( y = \frac{6 – 2x}{3} \) (dividing numerator and denominator by 2)
| x | 0 | 3 | -3 |
|---|---|---|---|
| y = (6 – 2x)/3 | 2 | 0 | 4 |
| Points | (0, 2) | (3, 0) | (-3, 4) |
Notice: Both equations give exactly the same points!
- Plot the line \( 2x + 3y = 6 \) passing through (0, 2), (3, 0), and (-3, 4)
- When you try to plot \( 4x + 6y = 12 \), you’ll get the exact same line
- Both lines overlap completely – they are the same line
- Every point on one line is also on the other line
- This represents infinitely many solutions
Example of coincident lines:
\( 2x + 3y = 6 \) and \( 4x + 6y = 12 \)
These lines overlap completely (they are the same line).
💡 Other Examples of Coincident Lines:
- \( x + y = 5 \) and \( 2x + 2y = 10 \)
- \( 3x – 2y = 6 \) and \( 6x – 4y = 12 \)
- \( y = 2x + 1 \) and \( 2y = 4x + 2 \)
📊 Summary of All Three Parts
Three Types of Line Relationships:
| Type | Condition | Example | Solutions |
|---|---|---|---|
| Intersecting | \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) | \( x – y + 1 = 0 \) \( 3x + 2y – 12 = 0 \) | One unique solution |
| Parallel | \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) | \( x + y = 4 \) \( x + y = 6 \) | No solution |
| Coincident | \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) | \( 2x + 3y = 6 \) \( 4x + 6y = 12 \) | Infinitely many solutions |
⚠️ Common Mistakes to Avoid
✅ Correct: Parallel lines never meet (different lines), coincident lines overlap completely (same line).
✅ Correct: Always calculate \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \) to determine the exact relationship.
✅ Correct: The triangle is formed by the two lines AND the x-axis. Find where each line crosses the x-axis (y = 0) and where the two lines intersect.
✅ Correct: For parallel lines, keep the coefficients of x and y the same but change the constant term.
💡 Key Points to Remember
- Intersecting Lines: Meet at exactly one point; different slopes
- Parallel Lines: Never meet; same slope, different y-intercepts
- Coincident Lines: Overlap completely; one is a multiple of the other
- Triangle Formation: Requires two intersecting lines and a third line (like x-axis or y-axis)
- X-axis Intersection: Set y = 0 to find where a line crosses the x-axis
- Y-axis Intersection: Set x = 0 to find where a line crosses the y-axis
- Creating Parallel Lines: Keep coefficients same, change constant term
- Creating Coincident Lines: Multiply entire equation by any non-zero constant
- Graphing Accuracy: Use at least 3 points per line for accuracy
📝 Practice Similar Problems
- Draw two lines that intersect at (3, 4)
- Create a pair of parallel lines with slope 2
- Write three different equations that represent the same line
- Question 7: Finding vertices of triangle formed by specific lines
Farhan Mansuri
M.Sc. Mathematics, B.Ed.
15+ Years Teaching Experience
Passionate about making mathematics easy and enjoyable for students.