For Which Value of k does

Question

For whic value of k does ;

kx+y=1

x+ky=1

Have one solution , no solution ,or infinitely many solution

Answer 1

A system of two equations in two variables can only have infinitely many solutions if one of the equations is linearly dependent on the other. In the case of two equations, that implies that one is proportional to the other. Thus, we must have k = c and ck = 1 for some constant c. This implies that k²=1, or k = +1 or -1. We could also have gotten this result from the determinant of the system.

For k=1, the two equations are the same. Any (x, 1-x) will solve them. This is an infinite set of solutions, the line x+y=1. For k = -1, we have x = y+1, but y = x+1, a contradiction. There are thus no solutions. For all other values, there will exist a single solution, namely y = 1/(k+1), x = 1/(k+1).