Question
Either find all integer solutions Or prove that there are no integer solutions for the equation 2l + Ty 147_
Either find all integer solutions Or prove that there are no integer solutions for the equation 2l + Ty 147_
Answers
Prove that there are no solutions in integers $x$ and $y$ to the equation $2 x^{2}+5 y^{2}=14 .$
In this problem were asked to identify three consecutive even numbers that add to 252. So what we're going to do is we're going to set up our algebraic equation, and we're going to start with X that I'll represent me first, even number. And then I need to add on my other two, even numbers that are in a row. So my second consecutive even number is going to be represented based off Next X plus, and I can't just go buy one that would give me an odd number. So I have to go up by two. And for my next even number, I'll have to go up by four in order to reach my next even number. So I have my three consecutive numbers represented by Variables, and they all add up to 200 52. So now I can go ahead and sell for X till I have three light terms on my left side with my Ex variables, X plus X Plus X gives me three X plus. My number values to plus four six equals 252. So now I'm going to deal with my six by subtracting six from both sides of my equation. That leaves me with three X equals 250 to attract six equals 200 46. And then I'm going to have to isolate my variable by dividing both sides by my coefficient of three. And I don't know what 246 divided by three is. So I'm going to go ahead and divide by a three off to the side here. So he goes into 24 8 times eight times three is told before subtract losing me with 06 Spring it down three goes into 62 times. And I know if I follow out the rest of my long division that I'm gonna end up with zero. An 82 is my answer. So I'm going to go ahead and write down X equals 82. So now that I have my first even number, I know that 80 to use my 1st 1 82 plus two has to be my 2nd 2nd consecutive and even into your that's gonna bring me in 84 and my next even number is going to be 86 x plus four. And those are our three consecutive imagers
According to given cushion here we have two numbers and some of the two numbers is given us. So most numbers is equal to seven and correct of the numbers is equal to 40. So let us suppose that the first number is X Let first number is next. Personal worries. That's so we can write. The second number is seven minus x. So second number second number is seven minus X Not the product of these two numbers has given us 40. So, Harry, tonight, ex Indo seven minus X is equal to 40. So here that means this is seven x minus X square and sifting the sporting to the left. Inside, we can write minus 14 equal to little. So this is minus X squared. Bless seven X minus 14 equal to little. Now, let us try to understand this where the solution is possible or not with the help of Bagram or with the help of graph of this. So the graph of dysfunction is so here. We can see that since the graph does not induce it, X axis does not interest it. Xx, sis, for that means we can see that this equation has no real solution. Cool. This aggression has no real solution. No solution. Notice it. Eah, So therefore no such numbers exist. So those we can see that it does no sense in numbers exist.
Question 50. Our problem is too wise. Where'd last seven? Why? Plus three equals zero, You know factor two. Why Water? Why? Plus I know my three has to go here cause do test three will be sick. And one times why will be what my And that led up to seven. Why equals zero? So I would have equal negative one or negative three.
Okay, Head. In this case, the solution of the car revolution is given to minus seven iota and two plus seven. Okay, so let's as you alphas equals two to minus seven iota and be dies. Equals stood toe plus seven. The question of coordination is simply even by excess squared minus four plus times off X plus. Alphabet is equal to X is equal to zero. So if I just put in here the question really simply excess squared minus two. Some of this too. So this seven I can sell some of this is simply forex. I never just product. So this is a plus two times two is simply for okay, this minus places minus. I'm just swallowing in here for the Air France purpose. So this is simply two times two is four less minuses. Minus 77 is 49. But this I swear, this minus one That begins. Plus one. So this is 49 plus four, which is nothing but for +33 So bless 53 is equal to zero. Is that acquired solution? OK,