Question

Question 2 (i) For which values of a and b will 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 solution?
3x+y=1
(2k-1)x + (k-1)y = 2k+1

Class 10 - Math - Pair of Linear Equations in Two Variables Page 62

Answer 1

The general form of a pair of linear equations

a1x + b1y + c1 = 0 , 

 a2x + b2y + c2 = 0

for infinite number of solutions  condition use : 

a 1/a 2 = b 1/b 2 =  c 1/c 2

For no solutions:

a 1/a 2 = b 1/b 2 ≠ c 1/c 2 

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Solution:

2x + 3y -7 = 0

(a – b)x + (a + b)y – (3a +b –2) = 0

On comparing with ax+by+c+0
a1=2, b1=3, c1=-7

a2= a-b , b2= a+b , c2= -(3a+b-2)

a1/a2 = 2/a–b = 1/2
b1/b2 = -7/a+b 
c1/c2 = -7/-(3a+b-2) = 7/(3a+b-2)

For infinitely many solutions,a1/a2 = b1/b2 = c1/c2

2/a–b = 3/a+b = -7/-(3a+b+2)

on taking I &II terms 

2/a–b = 3/a+b 

2(a+b) = 3(a-b)

2a+2b = 3a-3b

2a-3a=3a-2b

-a= -5b

a=5b ......................................(1)

on taking ii  &iii terms 

3/a+b = -7/-(3a+b+2)

3(3a+b+2) = 7(a+b)

9a+3b-6 = 7a+7b

9a - 7a +3b-7b =6

2a - 4b =6

2(a-2b) = 6

a-2b=3................................................(ii)

putting the value of a from eq 1 in eq ii

a-2b=3

5b-2b=3

3b=3

b=3/3

b=1

put the value of b in eq i

a=5b

a= 5×1

a=5

Hence the required values of a & b are 5 & 1

(ii) 

3x + y = 1

(2k –1)x + (k –1)y = 2k + 1

3x + y -1 = 0(2k –1)x + (k –1)y – (2k + 1) = 0

a1On comparing with ax+by+c+0

a1=3, b1=1, c1=-1

a2= 2k-1 , b2= k-1a+b , c2= -(2k+1)

a1/a2 = 3/2k-1
b1/b2 = 1/k-1 and
c1/c2 = -1/-2k-1 = 1/2k+1

For no solutions,
a1/a2 = b1/b2 ≠ c1/c2

3/2k-1 = 1/k-1 ≠ 1/2k+1

on taking I & ii terms

3/2k-1 = 1/k-1

3(k-1)=2k-1

3k-3 = 2k-1

3k -2k = – 1+3

k = 2

Hence, for k = 2, the given equation has no solution

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Hope this will help you....