General Solution Linear algebra

[Precursor]

Homework Statement
Find the general solution to the system:

$$ax+ by= 0$$
$$cx+ dy= 0$$

Consider the case when
$$ad- bc\neq 0$$

The attempt at a solution
I multiplied the first equation by "c" and the second equation by "a", and then I subtracted the two equations.

I got the following matrix:
$$0...ad-bc...0$$
$$ac...ad...0$$

Therefore, $$ad\neq bc$$. I got the general solution to be $$cx+ dy= 0$$ from the second row of the matrix. Is it right?

 

[vela]

Nope. The first line in your reduced matrix corresponds to the equation $$(ad-bc)y=0$$. Since $ad-bc\ne 0$, y must be zero. I'll let you figure out what x equals.

 

[Precursor]

Ok, since y= 0,

$$acx+ ady= 0$$
$$acx+ ad(0)= 0$$
$$acx= 0$$

So $$x= 0$$ too?

But that gives a general solution of $$0=0$$. Does it make it infinetly many solutions?

 

[Mark44]

You're making this harder than it needs to be. After discovering that y = 0, substitute that value in either of your original equations to solve for x.

There is only one solution to this system of equations.

 

[Precursor]

If I substitute 0 in for y into, say, the first equation, I still get x=0. So is that the general solution?

 

[Mark44]

The solution is x=0, y=0. That is the only solution.

 

[Dick]

Yes, that's the general solution. But just plugging it into ax+by=0 doesn't prove that. Suppose a=0??

 

[Precursor]

Yes, that's the general solution. But just plugging it into ax+by=0 doesn't prove that. Suppose a=0??

Is there a way, within the scope of this question, to determine whether a=0? Otherwise, I stick with x=0?

 

[Mark44]

Substitute y = 0 into both of your original equations. What do you get? If a = 0, as Dick mentioned, how does the condition that ad - bc != 0 affect things?

 

[Precursor]

If I substitute y=0 into both equations, I get ax=0 and cx=0. If a and c are both 0, then it won't satisfy the condition I stated in the problem. So either one of them is zero or neither. But I still can't see where you are going with this.

 

[Dick]

If I substitute y=0 into both equations, I get ax=0 and cx=0. If a and c are both 0, then it won't satisfy the condition I stated in the problem. So either one of them is zero or neither. But I still can't see where you are going with this.

You just got it. If a and c can't both be 0 then one of those equations tells you x=0.

 

[Precursor]

You just got it. If a and c can't both be 0 then one of those equations tells you x=0.

Ok, that makes sense. Thanks for the help all of you.

 

[vela]

The significance of this problem is that it tells you you don't have to actually solve the system of equations to see that x=0, y=0 is the only solution. You can just calculate ad-bc and see that it's non-zero.