Use the ratios a1a2, b1b2, c1c2, to determine whether the lines representing the following pairs of linear equations intersect at a point, are parallel or are coincident.
- (i) 5x − 4y + 8 = 0; 7x + 6y − 9 = 0
- (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- (iii) 6x − 3y + 10 = 0; 2x − y + 9 = 0
Step-by-step solution
Idea: a1a2 ≠ b1b2 ⇒ intersecting lines; a1a2 = b1b2 = c1c2 ⇒ coincident; a1a2 = b1b2 ≠ c1c2 ⇒ parallel. Keep the signs of the coefficients.
(i) 5x − 4y + 8 = 0; 7x + 6y − 9 = 0
- a1a2 = 57, b1b2 = −46 = −23.½ mark
- 57 ≠ −23, so the lines intersect at a point (unique solution).½ mark
(ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- a1a2 = 918 = 12, b1b2 = 36 = 12, c1c2 = 1224 = 12.½ mark
- All three ratios are equal, so the lines are coincident (infinitely many solutions).½ mark
(iii) 6x − 3y + 10 = 0; 2x − y + 9 = 0
- a1a2 = 62 = 3, b1b2 = −3−1 = 3, c1c2 = 109.½ mark
- a1a2 = b1b2 = 3 but c1c2 = 109 ≠ 3, so the lines are parallel (no solution).½ mark
Check: (iii) In slope form: y = 2x + 103 and y = 2x + 9: same slope 2, different intercepts ⇒ parallel ✓.
Answer to write in the exam
(i)
a1a2 = 57, b1b2 = −46 = −23
a1a2 ≠ b1b2
∴ The lines intersect at a point
(ii)
a1a2 = 918 = 12, b1b2 = 36 = 12, c1c2 = 1224 = 12
a1a2 = b1b2 = c1c2
∴ The lines are coincident
(iii)
a1a2 = 62 = 3, b1b2 = −3−1 = 3, c1c2 = 109
a1a2 = b1b2 ≠ c1c2
∴ The lines are parallel
Common mistakes that cost marks
- Dropping the minus sign in (i) and writing 46; here the conclusion is the same, but in other pairs the sign decides.
- In (iii), stopping after a1a2 = b1b2 and calling the lines coincident without checking c1c2.
How this can come in the exam
The lines 4x − 6y + 3 = 0 and 2x − 3y + 5 = 0 are
- intersecting
- parallel
- coincident
- the same as the axes
Show answer
(B) parallel
42 = −6−3 = 2 ≠ 35.
Try one yourself
Do 3x + 2y − 5 = 0 and 6x + 4y − 10 = 0 intersect, coincide or run parallel?
Show answer
36 = 24 = −5−10 = 12 ⇒ coincident.
More questions like this
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- Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
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- Here is a problem that was posed by Mahāvīrāchārya in Gaṇita sāra saṅgraha (c. 850 CE).
The price of 9 citrons and 7 fragrant wood-apples taken together is 107; and the price of 7 citrons and 9 fragrant wood-apples taken together is 101.
O mathematician, tell me quickly the price of each citron and of each fragrant wood-apple.
(Hint: The given problem can be modelled as
9x + 7y = 107
7x + 9y = 101
Can you figure out a way of solving these equations without directly using elimination or the substitution method? What is special in this pair of equations? The x and y coefficients are interchanged in the 2 equations.
What will happen if we add the pair of equations? What will happen if we subtract the pair of equations? Can the resulting equations be solved to find the values of x and y?) - 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of each pencil and pen.
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