What other ways can you split an equation into two?


by ainster31
Tags: equation, split
ainster31
ainster31 is offline
#1
Oct16-13, 10:42 AM
P: 154
Here are two ways:

$$(x-{ x }_{ 1 })({ x }-{ x }_{ 2 })=0\\ x-{ x }_{ 1 }=0\quad \quad \quad \quad x-{ x }_{ 2 }=0\\ \\$$$$ \\ { e }^{ x }({ c }_{ 1 }-3{ c }_{ 2 })+{ e }^{ -32x }({ c }_{ 5 }-{ c }_{ 4 })=0\\ { c }_{ 1 }-3{ c }_{ 2 }=0\quad \quad \quad \quad \quad { c }_{ 5 }-{ c }_{ 4 }=0$$

Any other ways?
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MrAnchovy
MrAnchovy is offline
#2
Oct16-13, 10:58 AM
P: 388
"Splitting an equation in two" is not in general a meaningful mathematical operation. Your second example is incorrect: can you see why? (clue: a = b = 0 is not the only solution to a + b = 0).
ainster31
ainster31 is offline
#3
Oct16-13, 11:24 AM
P: 154
Quote Quote by MrAnchovy View Post
"Splitting an equation in two" is not in general a meaningful mathematical operation.
Why not?

Quote Quote by MrAnchovy View Post
(clue: a = b = 0 is not the only solution to a + b = 0).
Hmmm... you're right. It's weird that my textbook teaches this technique.

What if it's an identity?

gopher_p
gopher_p is offline
#4
Oct16-13, 05:23 PM
P: 337

What other ways can you split an equation into two?


Quote Quote by ainster31 View Post
Here are two ways:

$$(x-{ x }_{ 1 })({ x }-{ x }_{ 2 })=0\\ x-{ x }_{ 1 }=0\quad \quad \quad \quad x-{ x }_{ 2 }=0\\ \\$$$$ \\ { e }^{ x }({ c }_{ 1 }-3{ c }_{ 2 })+{ e }^{ -32x }({ c }_{ 5 }-{ c }_{ 4 })=0\\ { c }_{ 1 }-3{ c }_{ 2 }=0\quad \quad \quad \quad \quad { c }_{ 5 }-{ c }_{ 4 }=0$$

Any other ways?
The first situation relies on the fact that if the product of two real numbers is zero, then one of those numbers must be zero.

The second situation is a statement about the linear independence of ##e^x## and ##e^{-32x}##; if ##ae^x+be^{-32x}=0## for all ##x## (i.e. ##ae^x+be^{-32x}## is the zero function), then ##a=b=0##. It's similar to the statement that if ##a_0+a_1 x+...+a_n x^n=0## for all ##x##, then ##a_0=a_1=...=a_n=0##.
MrAnchovy
MrAnchovy is offline
#5
Oct16-13, 06:07 PM
P: 388
Quote Quote by ainster31 View Post
Why not?
Because it does not in general yield anything useful.

Quote Quote by ainster31 View Post
Hmmm... you're right. It's weird that my textbook teaches this technique.
Perhaps the textbook is looking for solutions which are valid for all values of ## x ##?

Quote Quote by ainster31 View Post
What if it's an identity?
I don't understand what you mean.
Integral
Integral is offline
#6
Oct16-13, 06:54 PM
Mentor
Integral's Avatar
P: 7,292
Really this is not splitting an equation into two.

It is a way to find zeros of the original equation. Always keep in mind your goals, what are you trying to do. You are not splitting and equation you are looking for zeros.


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