General Solution Linear algebra

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  • #1
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Homework Statement
Find the general solution to the system:

[tex]ax+ by= 0[/tex]
[tex]cx+ dy= 0[/tex]

Consider the case when
[tex]ad- bc\neq 0[/tex]

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:
[tex]0......ad-bc......0[/tex]
[tex]ac......ad......0[/tex]

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

Answers and Replies

  • #2
vela
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Nope. The first line in your reduced matrix corresponds to the equation [tex](ad-bc)y=0[/tex]. Since [itex]ad-bc\ne 0[/itex], y must be zero. I'll let you figure out what x equals.
 
  • #3
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Ok, since y= 0,

[tex]acx+ ady= 0[/tex]
[tex]acx+ ad(0)= 0[/tex]
[tex]acx= 0[/tex]

So [tex]x= 0[/tex] too?

But that gives a general solution of [tex]0=0[/tex]. Does it make it infinetly many solutions?
 
  • #4
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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.
 
  • #5
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If I substitute 0 in for y into, say, the first equation, I still get x=0. So is that the general solution?
 
  • #6
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The solution is x=0, y=0. That is the only solution.
 
  • #7
Dick
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Yes, that's the general solution. But just plugging it into ax+by=0 doesn't prove that. Suppose a=0??
 
  • #8
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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?
 
  • #9
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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?
 
  • #10
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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.
 
  • #11
Dick
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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.
 
  • #12
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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.
 
  • #13
vela
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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.
 

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