Final answer:
To determine the value of k for which the system of linear equations has infinitely many solutions, we need to look at the coefficients of the equations. If the coefficients of the variables in the equations are proportional to each other, then the system will have infinitely many solutions. The value of k for which the system has infinitely many solutions is c) Any real number.
Step-by-step explanation:
To determine the value of k for which the system of linear equations has infinitely many solutions, we need to look at the coefficients of the equations. If the coefficients of the variables in the equations are proportional to each other, then the system will have infinitely many solutions. In other words, if the equations are dependent on each other, they will have infinitely many solutions.
For example, consider the system of equations:
2x + 3y = 5
kx + ky = 2
If we multiply the second equation by 2/3, we get:
(2/3)(kx + ky) = (2/3)(2)
(2/3)kx + (2/3)ky = 4/3
This equation is dependent on the first equation because the coefficients of the variables are proportional to each other. Therefore, if we choose k in such a way that the coefficients are proportional, the system will have infinitely many solutions.
Therefore, the value of k for which the system has infinitely many solutions is c) Any real number.