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10. ProveT-7 = XjJ?farsllsgts % artd(uz=t X} U Z foral seis %, andz li. PrsseX...

Question

10. ProveT-7 = XjJ?farsllsgts % artd(uz=t X} U Z foral seis %, andz li. PrsseX

10. ProveT-7 = XjJ?farsllsgts % artd (uz=t X} U Z foral seis %, andz li. PrsseX



Answers

Evaluate each expression when $x=-2, y=7,$ and $z=-3$ $$ x^{2}+x y+10 $$

Eight over Z -10 plus seven, oversee plus 10 equals five over z squared minus 100 so Z squared minus 100 Izzy plus 10 Z minus 10. So that's gonna be our common denominator. So when I multiply that the first term, the z minus tens reduce out and I'm left with eight times E plus 10. Mhm. When I multiply it by the second term I have seven times the minus 10 and when I multiply it by the five over Z plus n z minus 10, both of them cancel out and I'm left with just five. Next I distribute eight z plus 80 Plus seven, Z -70 equals five, Combine My Life Terms eight Z and seven Z is 15. Z. AT -70 -70 is plus 10 equals five. I want my variables on one side, my numbers on the other, so I want to subtract 10 here And that gives me 15, Z Minour equals -5, divide by 15 and reduce divide by five, Divide by five and we get negative 1/3.

Okay. Um, good day. Uh, so are we were the problem we're working on today as number three. And, um, as of the time of doing this, um, running this video Ah, the problem on the numerator website Page, um, is incorrectly labeled. It has ah, seven. It has 1/3 order differential equation. Um, saying Z to the third minus, um, or the third or the surf Z and the third derivative a Z minus six times the first derivative dizzy plus 10 z zero. In other words, the Z born in here is third order. It's 1/3 order instead of second order. But the book has a second order, and, um, it's just ah, mistake on their part. So hopefully they have changed it. I'm by the time you see this, but if you haven't, I can assure you it's a second order and not 1/3 order. Okay, So let down in mind, um, we go about solving it. Um, first we find the auxiliary equation, and in this case is just r squared minus six R plus 10. And if you're not clear on how to get that again, I would look back either at four, Section 4.3 or some of the earlier videos I did in 4.2, um, or even the book. And 4.2. I'll explain where that comes from. Um, but the route And after you found the auxiliary expression auxiliary equation, find the roots. And in this case, the roots are three plus or minus two. I and, um, here I I just used I'm trying to use that book notation, but I don't have an Alfa, so I just said elf a plus or minus or a plus or minus B I is once you have the Alfa Beta, then you can quickly see what the general solution is. And the form of the general solution, of course, is right here, Um, and in particular than we get this. Where are general solution? And, um, you should check, of course, that this is the correct general solution. But with all that's, uh, that's kind of that's kind of the whole problem. Um, the main thing is, if you're not sure about the auxiliary equation, please, just look that up. Um, I guess the other thing I could mention is that, uh, because the polynomial give you has complex roots. Um, I hope everyone knows how to find the complex routes. That would be something again to just look up. It's explained in the textbook. Pretty straightforward. Um, but I would just take a look if you're unclear. Okay, So with all that in mind, um, that, uh, that wraps his problem up. Thank you very much.

This problem. We're being asked to multiply To do this we're gonna use to distribute the property. So first we have to multiply Negative 10 by our first term. Negative six. Then we're gonna multiply Negative 10 by the second term in the prophecies, which is negative. Why? And then we're gonna have to multiply Negative 10 by the last term in the prophecies. Negative Z. Okay, so let's go ahead and multiply while negative 10 times negative. Six is positive. 60. Then we have negative 10 times Negative. Why? Which is positive? 10. Why? And lastly, we have negative 10 times negative Z, which is positive Tensy. And now, because none of these air like

In this video, we're gonna go through the Anson's question too benign. Chapter seven, part five to rest. If all of us to solve this initial value problem, that's a first, uh, so we want to use initiative to use look, less transforms. But it's only really gonna work if we have initial values, huh? To use equal zero. Okay, so less change the time. Very boob. So let how equal to t minus what? The towers under your time variable. It's just a, uh mm. Subtracting one from the time. So the derivatives with respect t are just gonna become derivatives with Spends time without changing anything. Um, so now, in terms of towel weaken right, this equation as, uh, everything is the same. Except we've just got tower here instead of t minus one on DDE. Uh, initial conditions in our violated zero rob someone. Okay, so now we can use Plus transforms, like, usual s o. We know what the the chancellor's insect intuited. Uh, so if big set capsules, that is the bus transport little that and it's minus minus. Want on this minus nine? Because five times questions. All of it is that crime, which is s times said, Plus well, minus six Zed. Well, big said then. 21. Last chance for 21 each. Tower 21. Over. Yes, Money. Okay, that's a no rearranging this four bigs that we've got 21. I guess that's what that affect all the, uh, terms and left inside that? No. Ah, it says a big said So it's gonna be s minus nine course. Fine. They went to divide by, um, over coefficients off the sets that said s squared was five s minus six. Okay, now let's try it. Sort this out. Let's most bottom bottom by X minus one. We're gonna get 21 minus that spineless fool. That's minus one on the denominator guys minus one. And then this X squared plus five minus six. Come fact arised the s cost six asked, minus one. So I made that square. Okay, so now we can use past affections again. I'm not gonna go through the details for this, but it's just that's standard procedure. Is that piquancy? Three of s minus one square, minus one of escorts. Six. And we know what the what? The investor class transforms off these. That's gonna tell us that Zed, as function of Tao is three e to the t times T minus eight months. Six t. Okay. Uh oh, hang on a minute. Thes tease. Thes t should have been toes, because now we can change back to the original variable T. So is that its function of tea is now three into the tea, minus one. Ties by T minus one minus mmm to the minus six. So you might as well. And that's the solution to this differential equation.


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