Question

Find the value of k for which the following system of equations have infinitely many solution:

x + (k + 1)y = 4 

(k + 1)x + 9y = 5k + 2

Answer 1

Given: x + (k + 1)y = 4, (k + 1)x + 9y = 5k + 2

Step-wise calculation:

1. For infinitely many solutions the two equations must be proportional:

`a_1/a_2 = b_1/b_2 = c_1/c_2`   ...(Coefficients of x, y, constants)

2. Here a₁ = 1, b₁ = k + 1, c₁ = 4 

a₂ = k + 1, b₂ = 9, c₂ = 5k + 2 

So `1/(k + 1) = (k + 1)/9 = 4/(5k + 2)`.

3. From `1/(k + 1) = (k + 1)/9` we get

(k + 1)² = 9 

⇒ k + 1 = ±3 

⇒ k = 2 or k = – 4

4. Test k = 2: `(k + 1)/9 = 3/9 = 1/3` 

And `4/(5k + 2) = 4/(10 + 2) = 4/12 = 1/3` 

⇒ consistent

Test k = – 4: `(k + 1)/9 = (-3)/9 = -1/3`

While `4/(5k + 2) = 4/(-20 + 2) = 4/(-18) = -2/9` 

⇒ not equal, so not acceptable

The system has infinitely many solutions only for k = 2.