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The values of a and b for which the following system of linear equations has an infinite number of solutions is:

2x + 3y = 7, (a + b)x + (2a - b)y = 21.

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a = 8 and b = 6

a = 5 and b = 1

a = 3 and b = 8

a = 5 and b = 6

answer is B.

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Detailed Solution

Given 2 equations 2x + 3y = 7 and (a + b)x + (2a - b)y = 21.
Simplifying the equations,
2x + 3y – 7 = 0
(a + b)x + (2a – b)y – 21 = 0
Comparing with standard equation,

a 1 = 2, b 1 = − 3, c 1 = − 7 a 2 = (a + b), b 2 = (2 a − b), c 2 = − 21

Now, condition for infinite solutions is

a 1 a 2 = b 1 b 2 = c 1 c 2

⇒ a 1 a 2 = 2 (a + b), b 1 b 2 = 3 (2 a − b), c 1 c 2 = − 7 − 21 ⇒ 2 (a + b) = 3 (2 a − b) = − 7 − 21

Consider case 1,

2 (a + b) = 721 ⇒ 7 a + 7 b = 42 ⇒ a + b = 6 ---- 1

Consider case 2,

3 (2 a − b) = − 7 − 21

⇒ 14 a − 7 b = 63 ⇒ 2 a − b = 9 ---- 2

Adding equation 1 and 2,

⇒

a + b + 2a – b = 6 + 9

⇒

3a = 15

⇒

a = 5
Now, the value of b is

⇒

a + b = 6

⇒

5 + b = 6

⇒

b = 6 – 5

⇒

b = 1
Therefore, a = 5 and b = 1.
Hence, option 2 is correct.

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