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# Repost but it's a image now

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6 https://imgur.com/a/JhMY5K2

I'm having trouble using LaTeX, uploading a photo from my computer, and making an account. Nothing works. The answer is 0 btw but I'm not sure how.   Feb 27, 2022
edited by Melody  Feb 27, 2022

#1
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I have played with it but I do not know how to solve it either. Feb 27, 2022
#2
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I sort of understand the solution now. Here is the answer.

https://imgur.com/a/HdMHSYG

Guest Feb 27, 2022
#3
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Here is the answer you have provided. It lost me after the first sentence ... Feb 27, 2022
#4
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I don't understand that answer, but algebraically it works out easily enough.

We need

$$\displaystyle \frac{a(ax+b)/(cx+d)+b}{c(ax+b)/(cx+d)+d}\equiv x.$$

Clearing the fraction within the fraction on the lhs,

$$\displaystyle \frac{a(ax+b)+b(cx+d)}{c(ax+b)+d(cx+d)}\equiv x.$$

For this to work we need three things to happen.

(1) the constant term on the top line should be zero, ie.     ab + bd = b(a + d) = 0,

(2) the x coefficient on the bottom line should be zero, ie.  ca + dc = c(a + d) = 0,

(3) The resulting fraction should cancel down to leave just x.

Applying (1) and (2) reduces the lhs to

$$\displaystyle \frac{x(a^{2}+bc)}{cb+d^{2}}$$

and this cancels down if

$$\displaystyle a^{2}=d^{2}.$$

$$\displaystyle a+d=0,( \text{ so }a=-d),$$ meets all of the requirements.

Feb 28, 2022
#5
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Thanks,

I should of thought of that but I didn't. Thanks for showing me  Melody  Feb 28, 2022
#6
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Thank you! That broke down easily. Guest Mar 1, 2022