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Question:

$$ \text{It is given that } y= \frac{3a+2}{2a-4} \text{and }x= \frac{a+3}{a+8} \\ $$

$$ \text{Express } y \text{ in terms of } x. $$

From using $x$ to solve for $a$, I discovered that $$ a = \frac{8x-3}{1-x} $$

Then I proceeded to substitute $a$ into $y$. I did this twice to ensure no mistakes are made, and my final answer for both was

$$ \frac{22x-7}{20x-10} $$

There's a problem, the correct answer is $$ \frac{7-22x}{10-20x} $$

This makes me want to cry, more so because I checked it twice and I was very careful about my working out, here it is:

$$ \frac{2+ 3(\frac{8x-3}{1-x})}{ 2(\frac{8x-3}{1-x}) -4 } $$

$$\rightarrow{}$$

$$ \frac{(\frac{2(1-x) + 3(8x-3)}{1-x})}{(\frac{-4(1-x)+2(8x-3)}{1-x})} $$

$$\rightarrow{}$$

$$ \frac{2(1-x) + 3(8x-3)}{1-x} * \frac{1-x}{-4(1-x)+2(8x-3)} $$

$$\rightarrow{}$$

$$ \frac{22x-7}{1-x} * \frac{1-x}{20x-10} $$

The $(1-x)$'s cancel out $$\rightarrow{}$$

$$ \frac{22x-7}{20x-10} $$

Can someone please tell me as to what I did incorrectly in the process? Thank you in advance!

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9  
You have actually done it perfectly. Multiply both numerator and denominator by $-1$ and you're done –  Marc Mar 14 at 6:24

2 Answers 2

up vote 11 down vote accepted

Your answer is $$\frac{22x-7}{20x-10}$$

And the books "correct" answer is $$\frac{7-22x}{10-20x}$$

Yes? Notice what happens when you multiply both the numerator and denominator in your answer by $-1$? You're very welcome.

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Ah! Thank you, I was wondering whether or not I should do that but I wasn't sure, thank you again! –  Samir Chahine Mar 14 at 6:33
2  
@SamirChahine you're welcome again. :D –  Sabyasachi Mar 14 at 6:43
3  
+1 for nonchalance. –  G. Bach Mar 14 at 16:30
2  
@G.Bach you mean +1 for sass? –  Sabyasachi Mar 14 at 16:33
    
They are easy to get mixed up, aren't they. –  G. Bach Mar 14 at 16:33

You got the correct answer. You just need to multiply both numerator and denominator by -1

$$\frac{22x-7}{20x-10} = \frac{-1}{-1} \times \frac{7-22x}{10-20x}$$

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1  
+1 This is a better answer than the one accepted since it makes explicit that OP original answer was correct. –  Taemyr Mar 14 at 9:29
1  
@Taemyr maybe. My answer was completely tongue in cheek. I just wanted to amuse myself. Apparently it amused the OP as well. –  Sabyasachi Mar 14 at 9:45
    
Is there a reason, other than just to match the "correct" answer, to do the multiply by -1/-1? Is there a rule or convention that says the "7 - 22x" form is preferred over the "22x - 7" form? –  Tim Mar 14 at 17:02
    
I don't think there is any rule but, I think there is a generally accepted (de facto?) convention of arranging terms in a polynomial order of decreasing powers. –  Brad S. Mar 14 at 17:10
    
@BradS. If you're working with polynomial rings then it's common to write the lowest degree terms first like here –  DanZimm Mar 14 at 18:47

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