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Pair of linear equations in two variables CBSE NCERT Solutions
(i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?
2x + 3y = 7
(a − b) x + (a + b) y = 3a + b – 2
(ii) For which value of k will the following pair of linear equations have no
3x + y = 1
(2k − 1) x + (k − 1) y = 2k + 1
(i) We compare equation 2x + 3y – 7 = 0 with a1x + b1y + c1 = 0 and (a − b) x + (a + b) y − 3a – b + 2 = 0 with a2x + b2y + c2 = 0.
We get a1 = 2, b1= 3, c1 = -7 and a2 = (a-b), b2 = (a+b), c2 = 2- b-3a.
Linear equations have infinitely many solutions if
⇒ 2a + 2b = 3a − 3b and 6 − 3b − 9a = −7a − 7b
⇒ a = 5b… (1) and −2a = −4b – 6… (2)
Putting (1) in (2), we get
−2 (5b) = −4b – 6
⇒ −10b + 4b = −6
⇒ −6b = –6 ⇒ b = 1
Putting the value of b in (1), we get
a = 5b = 5 (1) = 5
Therefore, a = 5 and b = 1.
(ii) We compare (3x + y – 1 = 0) with a1x + b1y + c1 = 0 and (2k − 1)x + (k − 1)y −2k – 1 = 0) with a2x + b2y + c2 = 0.
We get a1 = 3, b1= 1, c1 = -1 and a2 = (2k-1), b2 = (k-1), c2 = −2k – 1
Linear equations have no solution if .
⇒ 3 (k − 1) = 2k – 1
⇒ 3k – 3 = 2k − 1
⇒ k = 2
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