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
To determine the values of c for which there is a whole line of solutions with x=y=0, we need to find the values of c that make the system of equations consistent and dependent.
First, let's rewrite the equations in the form of Ax + By = 0:
1) cx + 2y = 0 --> cx + 2y - 0 = 0 --> cx + 2y = 0 --> x(1/c) + y(2) = 0
2) 2x + ay = 0 --> 2x + ay - 0 = 0 --> 2x + ay = 0 --> x(2) + y(a) = 0
To find the values of c for which there is a whole line of solutions (consistency and dependence), we need to check if the two equations are scalar multiples of each other.
For the equations to be scalar multiples, the coefficients of x and y in both equations must be proportional. Therefore, we can set the ratios of these coefficients equal to each other:
(1/c) / 2 = 2 / a
Cross-multiplying, we have:
(a/c) = 4
Now, we have a relationship between a and c: a/c = 4.
For a whole line of solutions, there shouldn't be any specific values for c for which only one solution is possible.
So, any value of c that satisfies a/c = 4 will result in a whole line of solutions. In other words, any value of c can be chosen as long as a is four times greater than c.
Conclusion:
The values of c for which there is a whole line of solutions with x=y=0 are any values of c such that a/c = 4.