Final answer:
To find the values of k for which the given pair of linear equations have infinitely many solutions, we need to set up the equations and determine the conditions. The value of k can be any real number, meaning there are infinitely many solutions for any value of k.
Step-by-step explanation:
To find the values of k for which the given pair of linear equations have infinitely many solutions, we need to set up the equations and determine the conditions.
Given equations:
kx + y = k² ...(1)
x + ky = 1 ...(2)
To have infinitely many solutions, the two equations must represent the same line. This implies that the slope and y-intercept of the two equations must be the same.
Comparing the equations, we have:
From equation (1): kx + y = k² → y = -kx + k² (3)
From equation (2): x + ky = 1 → x = 1 - ky (4)
Comparing equations (3) and (4), we can see that the slopes of both equations must be the same. Therefore, we have:
-k = -k (5)
From equation (5), we can conclude that the value of k can be any real number. Hence, there are infinitely many solutions for any value of k.