Rearranging Algebra to find Expression for Slope

**Apache** · July 30th 2009, 06:16 AM

Sorry about previous post,

The information I have is that I have to an expression for a slope of a curve.

I know that the slope of the curve is $\frac{-dy}{dx}$

$\frac{-dy}{dx} = \frac{-4x+4y+64}{4x-8y+32} = 0$

I reach this point, and I seem to fail to understand what I’m supposed to do, my method is that the

$\frac{-4x}{4x}$ cancels to leave $- x$

$\frac{4y}{-8y}$ gets $-0.5y$

$\frac{64}{32}$ gets $2$

Rearranging then gets you

$\frac{-dy}{dx} = 2-x-0.5y$

Which would be the expression for my slope?

**songoku** · July 30th 2009, 06:25 AM

Hi Apache

I reach this point, and I seem to fail to understand what I’m supposed to do, my method is that the

$\frac{-4x}{4x}$ cancels to leave $- x$

$\frac{4y}{-8y}$ gets $-0.5y$

$\frac{64}{32}$ gets $2$

Rearranging then gets you

$\frac{-dy}{dx} = 2-x-0.5y$

Sorry, you really have to review your algebra lesson.
That kind of division or simplification is absolutely wrong.....

You can't separate the terms like that.
You are right if the question turns out to be like this : $\frac{-4x}{4x} + \frac{4y}{-8y} + \frac{64}{32}$
But $\frac{-4x}{4x} + \frac{4y}{-8y} + \frac{64}{32}$ is completely different from $ \frac{-4x+4y+64}{4x-8y+32} $

Originally Posted by Apache

I know that the slope of the curve is $\frac{-dy}{dx}$

Almost right. The slope is $\frac{dy}{dx}$

Maybe you can post the whole question?

**Apache** · July 30th 2009, 07:03 AM

Im given a utilty function

$-2x^{2}+4xy-4y^{2}+64x+32y-14$

Use the total differential of the function to find an expression for the slope of any curve.

I followed the total differential steps and came to this,which im pretty sure is correct.

$\frac{dy}{dx}= \frac{-4x+4y+64}{4x - 8y +32}$

I then have to simplify this to get the expresion for the slope ?

All the terms are divisible by four, so if i did that id get,

$\frac{-x + y +16}{x - 2y +8}$

Would i be heading in the right direction ?

**bananaxxx** · July 30th 2009, 07:06 AM

Originally Posted by Apache

$\frac{-4x}{4x}$ cancels to leave $- x$

$\frac{4y}{-8y}$ gets $-0.5y$

$\frac{64}{32}$ gets $2$

Be careful.

$\frac{-4x}{4x}$ cancels to leave $- 1$

$\frac{4y}{-8y}$ gets $-0.5$

Think of it like
$\frac{-4x}{4x}=-1\times \frac{4}{4}\times \frac{x}{x}=-1$

**bananaxxx** · July 30th 2009, 07:11 AM

Originally Posted by Apache

$\frac{dy}{dx}= \frac{-4x+4y+64}{4x - 8y +32}$

Surely that IS your expression for the slope.

**Apache** · July 30th 2009, 07:28 AM

Well, the worked example im working from gives this,

$\frac{dy}{dx} = \frac{-20 - 4x - 2y}{16 - 2x - 2y}$

This then simplifys to,

$\frac{-10}{10} = -1$

So i guess ive got to try and do this with my equation, however it all goes a little hazy on how to do this.

**songoku** · July 30th 2009, 08:18 AM

Originally Posted by Apache

$-2x^{2}+4xy-4y^{2}+64x+32-14$

I suppose 32 should be 32 y

$\frac{dy}{dx}= \frac{-4x+4y+64}{4x - 8y +32}$

I think you lack of minus sign. It's expression for $-\frac{dy}{dx}$

Originally Posted by Apache

Well, the worked example im working from gives this,

$\frac{dy}{dx} = \frac{-20 - 4x - 2y}{16 - 2x - 2y}$

This then simplifys to,

$\frac{-10}{10} = -1$

So i guess ive got to try and do this with my equation, however it all goes a little hazy on how to do this.

Is this from the same question?

**Apache** · July 30th 2009, 08:38 AM

Yes, sorry it should be 32y.

No a different question but it follows the same format just uses different numbers, in the worked example case,

The utility function is

$U=20x + 16y - 2x^{2} - 2xy - y^{2}$

$ \frac{-dy}{dx} = \frac{-20 - 4x - 2y}{16 - 2x - 2y} $

Which on this example becomes,

$ \frac{-10}{10} = -1 $

**songoku** · July 30th 2009, 08:46 AM

If the utility function is : $U=20x + 16y - 2x^{2} - 2xy - y^{2}$

then, i think the slope should be :

$\frac{dy}{dx} = -(\frac{20 - 4x - 2y}{16 - 2x - 2y})$

To get $\frac{-10}{10} = -1$ , maybe you should substitute for a certain value of x and y

**Apache** · July 30th 2009, 09:33 AM

Sorry, I’ve got very confused by this and made a real hash of it, anyway ill work through both questions I posted to try and clear it up.

The worked example

$U=20x + 16y - 2x^{2} - 2xy - y^{2}$

This is the expression for the slope of any curve

$\frac{dy}{dx} = -(\frac{20 - 4x - 2y}{16 - 2x - 2y})$

Now, when x = 2 and y = 1, substituting those in gives,

$\frac{-10}{10} = -1$

In the case of my question

$U= -2x^{2} + 4xy - 4y^{2} + 64x + 32y -14$

Follow the total differential procedure gets me to,

$\frac{-dy}{dx} = \frac{-4x+4y+64}{4x-8y+32}$

Which is the expression for the slope

Now, when x = 20 and y = 10, substituting those in gives,

$\frac{-184}{32} = - 5.75$

I completely overlooked what i was meant to do as i got bogged down with trying to figure out the algebra division, when in truth it had nothing to do with it.

Thanks all, much appreciated.

**songoku** · July 30th 2009, 06:54 PM

Originally Posted by Apache

$\frac{-184}{32} = - 5.75$

It's better to check that once again ^^

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