Determine if the lines x + 2y − 4 = 0 and 2x + 4y − 12 = 0 intersect, coincide or are parallel to each other. Then verify your answer by graphing the equations.
Step-by-step solution
Idea: Compare the three ratios first; then draw each line through its two axis points and see that they never meet.
- a1a2 = 12, b1b2 = 24 = 12, c1c2 = −4−12 = 13.1 mark
- a1a2 = b1b2 but ≠ c1c2, so there is no solution: the lines are parallel.1 mark
- Points: x + 2y − 4 = 0 → R(0, 2), S(4, 0). 2x + 4y − 12 = 0 → P(0, 3), Q(6, 0).1 mark
- Drawing RS and PQ shows two lines with the same slope (−12) that never meet: they are parallel ✓.1 mark
Check: In slope form: y = −12x + 2 and y = −12x + 3: same slope, different intercepts ✓.
Answer to write in the exam
a1a2 = 12, b1b2 = 24 = 12, c1c2 = −4−12 = 13
a1a2 = b1b2 ≠ c1c2 ⇒ lines parallel
x + 2y − 4 = 0: (0, 2), (4, 0); 2x + 4y − 12 = 0: (0, 3), (6, 0)
Graph: the two lines do not meet
∴ The lines are parallel (no solution)
Common mistakes that cost marks
- Comparing only a1a2 and b1b2 and calling the lines coincident.
- Writing c1c2 = 412 with signs dropped; here both are negative so the ratio is still 13, but in other questions signs matter.
How this can come in the exam
The pair 3x − y + 2 = 0 and 6x − 2y + 4 = 0 represents
- intersecting lines
- parallel lines
- coincident lines
- perpendicular lines
Show answer
(C) coincident lines
36 = −1−2 = 24 = 12.
Without drawing, decide whether 2x − 5y + 1 = 0 and 4x − 10y + 7 = 0 intersect, coincide or are parallel.
Show answer
24 = −5−10 = 12 (1 mark), 17 ≠ 12 ⇒ parallel (1 mark).Try one yourself
Are x − y + 1 = 0 and 3x − 3y − 2 = 0 parallel? Check by ratios.
Show answer
13 = −1−3 ≠ 1−2 ⇒ yes, parallel.
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