Given the linear equation 2x + 3y − 8 = 0, write another linear equation in two variables so that the graphs of the pair formed represent (i) intersecting lines (ii) parallel lines (iii) coincident lines.
- (i) intersecting lines
- (ii) parallel lines
- (iii) coincident lines
Step-by-step solution
Idea: Intersecting: make a1a2 ≠ b1b2. Parallel: multiply the x– and y-coefficients by the same number but not the constant. Coincident: multiply everything by the same number.
(i) intersecting lines
- Take 3x − 2y − 7 = 0: 23 ≠ 3−2, so the lines intersect.1 mark
(ii) parallel lines
- Take 4x + 6y − 12 = 0: 24 = 36 = 12 but −8−12 = 23 ≠ 12, so the lines are parallel.1 mark
(iii) coincident lines
- Take 4x + 6y − 16 = 0 (2 × the given equation): 24 = 36 = −8−16 = 12, so the lines are coincident.1 mark
Check: (i) Solving 2x + 3y = 8 and 3x − 2y = 7: 13x = 37, so they do meet at one point (3713, 1013) ✓. (iii) (1, 2) lies on both 2 + 6 − 8 = 0 and 4 + 12 − 16 = 0 ✓.
Answer to write in the exam
(i)
3x − 2y − 7 = 0
a1a2 = 23 ≠ b1b2 = 3−2 ⇒ intersecting lines
(ii)
4x + 6y − 12 = 0
24 = 36 = 12 ≠ −8−12 = 23 ⇒ parallel lines
(iii)
4x + 6y − 16 = 0
24 = 36 = −8−16 = 12 ⇒ coincident lines
Common mistakes that cost marks
- For parallel lines, multiplying the constant as well (that gives coincident lines).
- For intersecting lines, choosing 2x + 3y − 5 = 0, which only changes the constant and gives parallel lines.
How this can come in the exam
Which line is parallel to x − 2y + 3 = 0?
- 2x − 4y + 6 = 0
- 2x − 4y + 1 = 0
- 2x + 4y + 1 = 0
- x + 2y = 3
Show answer
(B) 2x − 4y + 1 = 0
12 = −2−4 ≠ 31.
Try one yourself
Write a line coincident with x + 4y − 2 = 0 and one parallel to it.
Show answer
Coincident: 3x + 12y − 6 = 0; parallel: e.g. 3x + 12y + 1 = 0.
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O mathematician, tell me quickly the price of each citron and of each fragrant wood-apple.
(Hint: The given problem can be modelled as
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