Consider the system of equations kxy 2 2x 2y 4 For what

Consider the system of equations: kx+y = - 2 2x - 2y = 4. For what values of k does the system have: A unique solution? No solution? Infinitely many solutions?

Solution

Given Equations are kx + y = -2

2x - 2y = 4

comparing these two equations with a₁x + b₁y +c₁ = 0

a₂x + b₂y +c₂ = 0

here a₁ =k , b₁ =1, c₁ =2

a₂ =2, b₂ = -2, c₂= -4

since b₁/b₂ = c₁/c₂

a₁/a₂ must be equal to b₁/b₂

  k/2 = 1/-2

implies k = -1

Therefore given system of equations have infinitely many solutions for k = -1.