# How to express $z'(t)$ and $w'(t)$ in terms of $z(t)$ and $w(t)$?

I have these functions:

$x' (t) = −5x(t) + 2 y(t)$

$y' (t) = 2x(t) − 2y(t)$

where $x(0)=10$ and $y(0)=0$

I am also given these 2 functions:

$z(t) = x(t) + 2y(t)$

$w(t) = −2x(t) + y(t)$

First question is to express $z'(t)$ and $w'(t)$ in terms of $x'(t)$ and $y'(t)$

so:

$z'(t) = x'(t) + 2y'(t)$

$w'(t) = -2x'(t) + y'(t)$

Easy enough. I am then asked to express $z'(t)$ and $w'(t)$ in terms of $z(t)$ and $w(t)$, but I don't know how to do that!

Can I get some pointers? Thanks!

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For example $z'=x'+2y'=(-5x+2y)+2(2x-2y)=-x-2y=-(x+2y)=-z.$ – Git Gud Jun 15 '14 at 19:46
ugh that makes sense. I am so bad. Thanks ! – Kevin Jun 15 '14 at 20:11
You're welcome. If no one posts an answer, I suggest you answer the question yourself, so it doesn't come up as unanswered. – Git Gud Jun 15 '14 at 20:13

## 1 Answer

$$\textbf{Note: this is just tidying up question with an answer as provided by @GitGud.}\\ \textbf{So please refrain from voting on this question.}$$

$$z' = x' + 2y' = (-5x+2y) + 2(2x-2y) = -x -2y = -(x+2y) = -z$$

Similarly for $w$ we find:

$$w' = -2(-5x+2y) + (2x-2y) = 12x -6y = 6(2x - y) = -6w$$

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