# Find the value of k for which each of the following system of equations has infinitely many solutions : 2x + 3y = 2 (k + 2)x + (2k + 1)y - 2(k - 1) - Mathematics

Find the value of k for which each of the following system of equations has infinitely many solutions :

2x + 3y = 2

(k + 2)x + (2k + 1)y - 2(k - 1)

#### Solution

The given system of the equation may be written as

2x + 3y - 2 = 0

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

The system of equation is of the form

a_1x + b_1y + c_1 = 0

a_2x + b_2y + c_2 = 0

Where a_1 = 2, b_1 = 3, c_1 = -2

And, a_2 = k + 3, b_2 = (2k + 1), c_2 = -2(k -1)

For a unique solution, we must have

a_1/a_2 = b_1/b_2 = c_1/c_2

=> 2/(k + 2) = 3/(2k + 1) = (-2)/(-2(k -1))

=>2/(k +1) = 3/(2k +1) and 3/(2k +1) = 2/(2k - 1)

=> 2(2k + 1) = 3(k + 2) and 3(k-1) = (2k +1)

=> 4k +2  = 3k + 6 and 3k - 3 = 2k +1

=> 4k - 3k =  6- 2 and 3k - 3k = 1 +3

=> k = 4 and k =4

Hence, the given system of equations will have infinitely many solutions, if k = 4

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 3 Pair of Linear Equations in Two Variables
Exercise 3.5 | Q 14 | Page 73