these equation are certain to have the solution x=y=0 for which values of c is there a whole line of solutions? cx+2y=0 2x+ay=0

Answer 1

We can solve the system of equations using substitution:

cx + 2y = 0
2x + ay = 0

Solving for x in the second equation, we get:

x = -ay/2

Substituting this expression for x into the first equation, we get:

c(-ay/2) + 2y = 0

Simplifying, we get:

y(2-c/2)a = 0

Therefore, the system has a whole line of solutions (x = ky, for some constant k) if and only if a = 0 or c = 4/2 = 2.

If a = 0, then the second equation becomes 2x = 0, which implies x = 0, and so the system has a whole line of solutions (y can take any value).

If c = 2, then the first equation becomes 2x + 2y = 0, which implies x = -y, and so the system has a whole line of solutions (y can take any value, and x is determined by -y).