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Start with [x1, Xz]"-[2.0, 0.5] and perform only_one Newton'$ iteration to find [X1, X2]" for the following system of nonlinear equations_ 0 = X12 - ...

Question

Start with [x1, Xz]"-[2.0, 0.5] and perform only_one Newton'$ iteration to find [X1, X2]" for the following system of nonlinear equations_ 0 = X12 - 2X1 - X2 0.5 0 = 4x22 - 4 + X12

Start with [x1, Xz]"-[2.0, 0.5] and perform only_one Newton'$ iteration to find [X1, X2]" for the following system of nonlinear equations_ 0 = X12 - 2X1 - X2 0.5 0 = 4x22 - 4 + X12



Answers

Find all solutions to the system of linear equations.
$\begin{aligned} x_{1}+2 x_{2} &=0 \\ 2 x_{1}+5 x_{2} &=1 \end{aligned}$

The first thing we're gonna do and we solve this problem is we're going to take the second equation and we're going to solve it for the variable Y. So that we have something to substitute. So I'm taking this equation right here, I'm gonna move five Y to the other side by adding it to both sides. So now I would have 12 X equals five Y. And then I'm gonna divide both sides by five. So 12 5th x would equal y. So now we're gonna substitute 12 5th X in place of Y. And the other equation. So that would turn that into X squared plus 12 5th X Quantity squared is equal to 169. So I replaced the letter Y with what it's equal to right there and now I'm gonna square what's in parentheses right here, that's going to get squared. So X squared stays the same at the beginning. 12 squared is 144 5 squared is 25 then X squared is X squared Still equal to 1 69. To solve this equation, I'm gonna multiply everything by 25. So they don't our so they can get rid of that fraction in the middle there. So 25 times X squared at the beginning is 25 X squared. The 25's would cancel their leaving us with 144 x squared. And the other side 169 times 25 is 4225. Yeah, combine like terms on the left and we have 169 X squared equals 4225 divide both sides by 169. And when you do that you get mhm X squared equals 25. Take the square root of both sides and you get x equals plus or -5. I'm gonna come back over here and use this linear equation to figure out my Y. Values. So I'm gonna substitute now in place of X. I'm gonna first put five. So that would be 12 times five equals five times Y. So that's 60 Divide both sides by five and you'll just kind of see right here that Y equals 12. So one answer is the ordered pair 5 12. The other one is what you get when you put the negative five end. So that would be 12 X. Or not. 12 back. We're gonna substitute 12 times negative five equals five. Why divide both sides by five? And now I have negative one. So negative one times 12 is negative 12 equals Y. So your second and final answer is the ordered pair negative five negative 12.

This problem, it is asking us to solve the system of nonlinear equations because our first equation is a quadratic since it has X squared, so because we have a quadratic and linear, this equation might may have the most to solutions because a problem and the line can intersect at most two points. So you can solve this system of equations either by the elimination method or the substitution method, but because my coefficients in front of why or the same number one and one except one is positive, one is negative. I can just go ahead and solve this equation by elimination because when I add those two equations together, one plus negative one becomes zero, so your wife is gone. So now you just have X square plus X equals to two. So now this is a quadratic equation. And to solve any quadratic equation you must set equal to zero. So I'm going to move the to to the other side. So now I have X squared plus x minus two equals to zero. So now you can solve this quadratic equation either by factoring or by using the quadratic formula. And because this quadratic equation is fact herbal, then we should solve this by factoring where my constant is negative 21 times negative for the number here in front of expo is one. So that's why you're just looking at negative two and you're finding two numbers that will multiply giving negative tube but add give you one which is the number in front of X. So the two number that will multiply gave you two we know is one and two, but we have to decide where we should put that negative one. So we know that negative one has to be in front of one because negative one plus two gives you one. So therefore the factor form for this quadratic equation is going to be x minus one times X plus two equals to zero. And if you're not sure, you can always foil combine like terms and say the same thing as the original quadratic equation. So now because I because I have written this equation in the factor form now to solve for X, you just say each factor equal to zero. So I have X -1 equals to zero exports to equals to zero. And now you're gonna solve for X. So you're gonna add one, so X equals to one and then the other one you're going to subtract two. So X equals negative two. So those are my two possible X values. And so to find the Y values, you can plug those X values into one of these original equations. And I think it's easier if you just plug into this linear equation. So my original linear equation was X minus Y equals to zero. So I'm just gonna plug X into one year, so one minus Y equals to zero. And to solve for why, I can just add one on both sides. So my first possible solution Is one is X, and one is the why. And then for the second exhale, you do the same thing. X months, Y equals zero, is gonna become negative, two minus Y equals zero. And again I'm going to add one on both sides to software wide, So therefore y equals two negative chip. So therefore my second solution for this system of equations is going to be X is negative two and why is also negative two. So these are the two solutions for this system of nonlinear equations.

No considered giving system of nonlinear equations over here. The first one is two, X, Y plus one equal to zero, and the second one is X plus 16, Y is equal to two. So these are the two equations given to us and we are supposed to find out the solution for this given system. So now you can either use elimination method or the substitution method. I'll be using the substitution method over here. So for that I'll be deriving the value of X in terms of vibe from equation to so X will be equal to minus 16. Why? Let's say this is equation three. Now, okay, now just put the value of eggs from equation three. Put eggs from equation three in a question, why? What? You'll get two times off X. That is two minus 16. Y times Y plus one is equal to zero. As you can see, I'm just substituting the value of X right now. Just open up the brackets. So we'll get four. Why minus 32? Y squared plus one is equal to zero right on. Uh if I just want to write it in standard quadratic form, that will be 32 Y squared minus four, Y minus one equal to zero. Now this is just a quadratic equation and you need to find out the factors for which you can either use the quadratic formula or you can use the characterization method. So on using the Factory isn't Factory ization method, like splitting up the middle to what we did, 32 wives where minus eight Y plus four, five minus one equal to zero. That will give you the factors as eight, Y plus one and the other one will be four by minus one, which is equal to zero. Hence the factors will be wise equal to one upon four or why will be equal to -1 upon it. So now we've got two values of I will substitute both. The values in equation three now. So let's say substitute by is equal to one upon four in equation three, what was the question three? If you remember guys that was excessive quarter to minus 16? Y okay So let's put it in equation three. So excellent. Equal to -16? Why? That is one upon four. Right. So on solving this what to minus four? That is a call to minus two. That means the solution will be minus two comma one upon for in the same anil substitute why is equal to minus one upon eight again in equation three, so X will be equal to two minus 16 times by that is minus one upon eight again to solve this, what we'll get to plus two, that is four. So the solution over Hillary for common minus one upon eight. Now, when you just combine both the solutions, the complete Solutions said for the given system will be minus two. Command one upon four, and the other one will be four. Commandment in this one upon it, so this is the complete solution set for the given system.

Now here, in this question, we've been giving us a talk to nonlinear equations over here. So the first equation that has been given to us is why is equal to x square plus four X. And the second equation is two, X -Y is going to -8. So we have to find out the solution to this given system of nonlinear equations over here. So let's see. This is a question one and this one is equation too. So I'll be using the substitution method over here. So that means what I'm going to do is I'm simply just going to substitute the value of wife From equation one in equation two. Okay, so I'm doing that. What I'll get simply uh, two X minus X squared Plus four X is equal to minus it right now, I'll just open up the brackets and I'll rearrange the domes. So I'm doing that, I'll get X square plus two X minus eight is called +20 Now, this is just a contract equation over here. If you want you can use the quadratic formula, you can split up the middle toe. I just wouldn't want to find out the factors to just give it a quadratic equation over here. So the factors or will be X -2 and the other one will be expressed for over here to the quadratic equation. Now from this, I can say that X is equal to two or X is equal to minus for right now we've got two values of X and we'll just put this value of X in equation what? So just put X is equal to two in equation one. What was the question one? That was why is it going to excess square plus four X. And similarly put excess cooled to -4? Also in equation one. So I'm doing that. What we'll get simply, why is it called to to swear that is four plus eight? That will be 12. That means one solution will be to home a 12. And when you put excess equal to minus four in question, one will get why is equal to 16 minus 16, that is equal to zero. That means the other solution will be minus four comma zero over here. Hence I can see that the solution to the given system will be two comma 12 and the other one is minus four comma zero. So this will be the solution. So as you can see, we found out the solution to the given system of non in equation by using the substitution method over you. If you want, you can use the elimination method aspect.


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