Question

Determine the values of a and b for which the following system of equations has infinite solutions.

2x – (a – 4)y = 2b + 1 , 4x – (a – 1)y = 5b – 1

विकल्प

• a = 7 and b = 3

• a = 7 and b = 1

• a = 5 and b = 2

• a = 2 and b = 7

Answer 1

a = 7 and b = 3

Explanation:

Considering 2x – (a – 4)y = 2b + 1

Comparing with a₁x + b₁y = c₁

We get, a₁ = 2, b₁ = – (a – 4) and c₁ = 2b + 1

Now, considering 4x – (a – 1)y = 5b – 1

Comparing with a₂x + b₂y = c₂

We get, a₂ = 4, b₂ = – (a – 1), and c₂ = 5b – 1

For infinite numbers of solution,

`a_1/a_2 = b_1/b_2 = c_1/c_2`

∴ `2/4 = (-(a - 4))/(-(a - 1)) = (2b + 1)/(5b  - 1)`

Considering `(a - 4)/(a - 1) = 1/2`

⇒ 2a – 8 = a – 1, ∴ a = 7

And now for 'b'

`(2b + 1)/(5b - 1) = 1/2`

⇒ 4b + 2 = 5b – 1

∴ b = 3

∴ The given system of equation will have infinitely many solution if a = 7 and b = 3.