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$53-56$ Find $x$ and $y$ in terms of $a$ and $b$.
$$
\left\{\begin{aligned} a x+b y &=0 \\ a^{2} x+b^{2} y &=1 \end{aligned} \quad(a \neq 0, b \neq 0, a \neq b)\right.
$$

In this problem. We have toe valve eight. Using the result off Question 52. Now we have given limit edge approaches. Zero co signed three H, minus one over at square. Now, leg three edges equals toe Do. Therefore, we have edges equals toe. Be over three. Therefore, we will write the given function as limit the approaches zero cosigned D minus one over the over three. Whole square, therefore began. Limit be approaches zero nine. Multiply or signed P minus one over P square. Now for the simplifying it. We get nine limit. He approaches zero minus one minus society over. Be square. No, for the simplifying it figure minus nine limit. He approaches 01 miners assigned T over P square now in question. Toe to you know that limit H approaches 01 minus four. Signed edge over. Red Square is it was the one over to that before we can write this ass minus nine. Medical. I've won over to before. We get minus nine over to hence limit H approaches zero. Go sign three H minus one over. Ed Square is equals toe minus 9/2. So the final solution is this one

Okay, so we're asked. It's off the system using a graphing calculator. So what we have to do first, we solved for each equation, pushing one and two in terms of Why so starting with equation one. It's a check for ad for 35 acts on both sides. Get nine. Want to know why is he could do 435 x and dividing by 912 on both sides. Forget for $35 5 91 too, Which is why is equal to zero point for eight X. Okay, now working with the regime to I have, I'm going to do subtract negative earth object 1 32 taxable sites get 4555 Why you come to nine? I mean, four minus one Dory to X. Now dividing my 4 55 above science, I get why you go to 994 divided by 4 35 which is 2.1 hey minus 13 to 45 just zero point to nine x. Okay, now, since I have buried two equations in terms of why I plug it into my graphing calculator and I get this so this There's two lines whose sorry about that. The Intersect at one point just here, you know I have to do now is round two thio second. But you do this in those places. Which is 2.78 common, 1.3.

Hello. I hope you're doing well. We have our system of equations here. We're going to try and solve this using the elimination. So, in trying, eliminate or variable, why So to do that, we're going to multiply this first equation I a This whole second equation by minus V. This is starting with our first equation in distributing the A eight times a axes, a squared ice, a times B y plus a B Y. That's equal to eight times one, which is equal to a moving on to our second equation and distributing the minus A B. We have minus three times be X is minus B squared acts. We have minus a few times a wise minus a B Y. That's equal to minus three times one, which is minus feet now, adding these two equations together right here we have a B Y minus a B Y C skin salon. Good a zero. So we're left with a squared X minus B squared X, which is a squared minus B squared X that's equal a minus. Bi fighting both sides of our equation by a squared minus B squared. We end up with end up with X is equal to a minus bi over a squared minus peace with they scored minus B squared is the difference of squares That's equals a plus. B sounds a minus. Bi in our numerator is still a minus bi, so these a modest fees can cancel on We're left with X is equal to one over a plus B. All right, so now we want to try and figure out what why value corresponds to this. Expel you. So working in do that by rewriting our first equation here, we can plug in our value for X one of her A plus B so eight times. So our axes, one of her a plus B plus to be why is equal to one. This is equal to a over a plus. B plus view. Y is equal to one solving for why we're going to subtract ever a plus B from both sides of our equation. When we do that, we end up with B Y is equal to a one minus a over a plus city and dividing both sides of our equation by B, we end up with why is equal to one over B minus a over B times a velocity. So let's get a common denominator now be times a policy. So to do that. But this one overview we're going to multiply, but the numerator and denominator by a plus B we end up with a plus B Overbey times a plus B minus a Overbey times a plus. So since we have a common denominator of be tense a plus B, we can just subtract this a plus pianist se se plus b minus a just gives us to be So we end up be Overbey times a plus B. These species cancel out. So we end up with why is equal to one over a plus B. I remember from before we ended determined that X is also equal to 1/8. Plus, some things are final answer. Our solution to the system of equations is going to be X. Why? Or one over a plus B one over a possible. This is our final answer right here. All right, well, thanks. And I hope that hopes

The theme in this problem is that the limit as H approaches zero of F of X, passage minus F of X all over. H is our definition for the derivative of X. So as you look at this problem and they tell you you're doing the in limit has h approaches zero and they give you um five And then instead of exit, right 1/2 plus h. Uh to the 8th power -5 times one half To the 8th power All over. H. You're finding f crime of one half. Um And let me circle in blue, you see how F of X and F. Primer right here. Um So this would be Fx and then this would be F prime of X. So the function for F of X Is equal to five times x to the 8th power. And then that derivative since we've learned in this section, would be eight times five. That's how we do this power rule To the 7th power. Um And then from there, you have to plug in one half, and for X. uh which means it's the same thing as two to the 7th power uh in the denominator. So 248 16 32 64, Which would actually make sense to divide by two a couple of times, like 20/64, Or 10/32 Or 5/16.