<s>[INST] All fractions which can be written simultaneously in the forms $\frac{7k-5}{5k-3}$ and $\frac{6l-1}{4l-3}$ Find all fractions which can be written simultaneously in the forms $\frac{7k-5}{5k-3}$ and $\frac{6l-1}{4l-3}$
for some integers $k,l$.

Please check my answer and tell me is correct or not....
$$\frac{43}{31},\frac{31}{27},1,\frac{55}{39},\frac{5}{3},\frac{61}{43},\frac{19}{13},\frac{13}{9}$$

 [/INST] $$\frac{7k-5}{5k-3}=\frac{6l-1}{4l-3}$$
$$28kl-20l-21k+15=30kl-18l-5k+3$$
$$2kl+2l+16k-12=0$$
$$kl+l+8k-6=0$$
either:
$$l(1+k)=2(3-4k)$$
so
$$l=2\frac{3-4k}{1+k}$$
or:
$$k(l+8)=6-l$$
so
$$k=\frac{6-l}{l+8}$$
Let's go with this second one to complement the other answer.
Then $$k=-\frac{l-6}{l+8}=-\left(\frac{l+8-14}{l+8}\right)=-\left(1-\frac{14}{l+8}\right)$$
We want $l+8=\pm(1,2,7,14)$
So $l=-7,-9,-6,-10,-1,-15,6,-22$ or more nicely ordered $$l=-22,-15,-10,-9,-7,-6,-1,6$$
The pairs are then $(l,k)=$ $(-22,-2)$, $(-15,-3)$, $(-10,-8)$, $(-9,-15)$,  $(-7,13)$, $(-6,6)$, $(-1,1)$, $(-22,-2)$, $(6,0)$, 
or that the fractions are
$$\frac{19}{13},\frac{13}{9},\frac{61}{43},\frac{55}{39},\frac{43}{31},\frac{37}{27},1,\frac{5}{3}.$$
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