Question

Find the value of k for which the system of linear equations has an infinite number of solutions.
2x + 3y – 7 = 0,
(k – 1)x + (k + 2)y=3k

Answer 1

The given equations are
2x + 3y – 7 = 0                 ……(i)
(k – 1)x + (k + 2)y – 3k = 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 = -7, a_2 = k – 1, b_2 = k + 2 and c_2 = -3k`
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/(k−1) = 3/(k+2) = (−7)/(−3k)`
`⇒ 2/(k−1) = 3/(k+2) , 3/(k+2) =( −7)/(−3k) and 2/(k−1 )= (−7)/(−3k)`
⇒ 2(k + 2) = 3(k – 1), 9k = 7k + 14 and 6k = 7k – 7
⇒ k = 7, k = 7 and k = 7
Hence, k = 7.