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What other ways can you split an equation into two?

by ainster31
Tags: equation, split
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ainster31
#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
#2
Oct16-13, 10:58 AM
P: 441
"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
#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
#4
Oct16-13, 05:23 PM
P: 432
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
#5
Oct16-13, 06:07 PM
P: 441
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
#6
Oct16-13, 06:54 PM
Mentor
Integral's Avatar
P: 7,315
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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