Find the value of k for which the following system of linear equations has infinite solutions: x + (k + 1) y = 5 and (k + 1)x + 9y = 8k – 1

Comparing the above equations with general form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

Then we get:

a1 = 1, b1 = k + 1, c1 = 5

a2 = k + 1, b2 = 9, c2 = 8k – 1

We know that an infinite number of solutions can exist for a pair of linear equations.

a1/a2 = b1/b2 = c1/c2

1/(k + 1) = (k + 1)/9 = 5/(8k – 1) …. (3)

Consider the first and second terms in(3)

1/(k + 1) = (k + 1)/9

On rearranging we get:

(k + 1)2 = 9

The value of k is:

k + 1 = 3, k + 1 = -3

k = 2, k = -4

Take the first and last (3)

1/(k + 1) = 5/(8k – 1)

On rearranging we get:

5(k + 1) = 8k – 1

5k + 5 = 8k – 1

3k = 6

The value of k is: k = 2

Thus, the given system of equations has an infinite number of solutions when k = 2.

Hence, the correct answer is option C.

Summary:

Find the value of k for which the following system of linear equations has infinite solutions: x + (k + 1) y = 5 and (k + 1)x + 9y = 8k – 1

The value of k for which the following system of linear equations has infinite solutions x + (k + 1)y = 5, (k + 1)x + 9y = 8k – 1 is 2. First-order equations include linear equations. In the coordinate system, linear equations are defined for lines.