Question

Find the value of k for which the system of linear equations has an infinite number of solutions.
2x + 3y=9,
6x + (k – 2)y =(3k – 2

Answer 1

The given pair of linear equations are
2x + 3y – 9 = 0                            ……(i)
6x + (k – 2)y – (3k – 2) = 0           ……(ii)
Which is of the form `a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0`, where
`a_1 = 2, b_1 = 3, c_1 = -9, a_2 = 6, b_2 = k – 2 and c_2 = -(3k – 2)`
For the given pair of linear equations to have infinitely many solutions, we must have
`(a_1)/(a_2) = (b_1)/(b_2) ≠ (c_1)/(c_2)`
`⇒ 2/6 = 3/(k−2) ≠ (−9)/(−(3k−2))`
`⇒ 2/6 = 3/(k−2) , 3/(k−2) ≠ (−9)/(−(3k−2))`
`⇒ k = 11, 3/(k−2) ≠ 9/((3k−2))`
⇒ k = 11, 3(3k – 2) ≠ 9(k – 2)
⇒ k = 11, 1 ≠ 3 (true)
Hence, k = 11.