Question

Find the value of k for which the system of linear equations has an infinite number of solutions.
10x + 5y – (k – 5) = 0,
20x + 10y – k = 0.

Answer 1

The given pair of linear equations are
10x + 5y – (k – 5) = 0                  ……(i)
20x + 10y – k = 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 = 10, b_1 = 5, c_1 = -(k – 5), a_2 = 20, b_2 = 10 and c_2 = -k`
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)`
`⇒ 10/20 = 5/10 = (−(k−5))/(−k)`
`⇒ 1/2 = (k−5)/k`
⇒ 2k – 10 = k ⇒ k = 10
Hence, k = 10.