Question

Show that the system of equations

2x – 3y = 5, 6x – 9y = 15

has an infinite number of solutions.

Answer 1

The given system of equations:

2x – 3y = 5

⇒ 2x – 3y – 5 = 0   ...(i)

6x – 9y = 15

⇒ 6x – 9y – 15 = 0   ...(ii)

These equations are of the following forms:

a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0

Here, a₁ = 2, b₁ = –3, c₁ = –5 and a₂ = 6, b₂ = –9, c₂ = –15

∴ `(a_1)/(a_2) = 2/6 = 1/3,(b_1)/(b_2) = (−3)/(−9) = 1/3` and `(c_1)/(c_2 )= (−5)/(−15) = 1/3`

Thus, `(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`

Hence, the given system of equations has an infinite number of solutions.