Find all solutions of the linear congruence $3 x-7 y=11(mod 13)$....

Find all solutions of the system of equations.
$\left\{\begin{array}{c}{\frac{2}{x}-\frac{3}{y}=1} \\ {-\frac{4}{x}+\frac{7}{y}=1}\end{array}\right.$

Okay, so we want to decide if these images are important to three more. Seven. So although we need to do is just simply taking model them. So 37 Most seven. So then I'll use the same trick as I did before. So this you subtract seven times by floor off 37 divided by seven. So this number right here is Why so the 37 minus by 35 which is equivalent to two models. So? So this is not a girl be 66 mother seven. So once again, 66 minus by seven times by floor, 66 by seven. So this number here gives you nine. So it will be 66 minus by seven times nine gives you 63. So which is equivalent to three more, sir? So this is true. Now, see? Negative. 17 mold seven. Well, negative. 17 mile, despite seven times by floor. Negative. 17.7 this summer. Right here is negative too. So it would be negative. 72 negatives make a plus plus by seven times to give you 14 which is negative. Three. So this is equivalent to four more seven. And therefore it is not now. Today, many to 67 mold seven. So you could use the usual way, as we did here. But then I'm gonna use this trick right here. We know that 66 66 is it is equal to three more seven. So if we put negative on both sides, so negative 66 is equivalent to negative three. Mother seven now is the track one from both sides. So negative 67 is equivalent to negative for more. Seven. But, hey, we also know that negative for months seven is equal to three. Well, so answer therefore, this is true.

I get. So we want to solve the concurrence relationship to excess equipment to seven mud 17. So it just all this we used the investor to invest is equivalent to nine more 17 now solving this All we did is we take will not apply to invest on both sides. Did you invest time by two ex is equivalent Thio seven type of two in verse More 17 chain gets his equal tonight with the two of us is castles So exits the equipment Thio £9.7 which is 63 63 model 17 No, we have first we want to find 63. You got about 70 Florida's. So this is it. 23 And we have 61 have 17 plants were three is equal to just have 63 is even to three tons of 17 plus by Flo 00 So heads 63 months 17 This one cattle leaving you tell that this is just equal were equivalent to 12 Move 17

Okay, so this question we want to solve the condolence. 15 X squared, plus by 19 x is equivalent to fire my love. So the first thing we'll do is we'll remove the five. I want to make a quadratic version X where, but by 19 explains by five, is equivalent to zero one. You know, first thing that we have to do then is fact arises. So if you want a fact the rights issue can I just give you the answer, which is just three eggs, plus by two times by by ex Spy three is equivalent to zero, not you. So in the same sense that if you have eight times by B, is it zero? This implies that a easy to zero or B zero you do the same thing here, and you say that this implies three X plus by shoot is equivalent to zero more 11 or why X Plus three is equivalent to reserve more 11. Now what we'll do is walk the track to to the other side, so three eggs. Is it good less to minus Pikachu. More 11 and modest, too, is equivalent to 11 pluck even 11 11. My spot, too. So that gives you a night. So nine modern 11 that at this point we can't divide three because the greatest common divider between 11 and three is one that we can do by three exes equivalent to three morning. So now we want to solve this one. So subtracting three ever so by ecstasy problems too negative. Three more 11 11 months, three cues. You ate more. 11. Now we want to find the invest off five months. 11. So if we think about two numbers that give us one which has 5 11 you just think about it. It's going to be 10 and 11 now. 10 mod 11 is it gives you months one. So what does this mean? Well, it means that needs to be in minus 10 11 to give us a plus one. So then the endless off five. So why type us a number? A. We want this to be minus 10. So is it my response to that? We must fight everything by miles to we have Ecstasy equivalent to negative or 16 more. 11. Now sitting. What? 11 2 alliances This soon we should give you cease. Well, you know. So hence the result is either three more 11 or six of them. And if we write that career, Ecstasy current three More 11 or X is equivalent to six more beer.

Okay, so in this question, we want to solve this system off Concours. So pressing that will do is we fully right At this model. The X is equal to three plus six a where a is some interview now what we'll do is we'll have well, substitute this into this. More question. So three plus six A is equivalent before more. Seven. Subtract Street six is equivalent to one. Oh, we'll soon Now that's six months. Seventies mice one. So this is making a is equivalent to one well seven. And so a is equipment to minus one, which is just six months in. So what this means is that a is equal to six plus five seven. Let's say B for the something to job. And if we substitute this into this equation, we have X is an equal to three plus by six and six plus seven Big. This is three plus two countries is 36. Accept 70 42. That's what for you to be. There's express treated you 39 Well, that's like for you to be. And by definition of mud, this is just equal to 39. One for you too.