In Exercises 1–18, solve each system by the substitution method. ...

Question

In Exercises 1–18, solve each system by the substitution method. xy=3, x^2+y^2=10

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve the given system of equation using the substitution method. Okay. And the equations were given our X Y equals two and x squared plus Y squared is equal to five. Okay, So let's go ahead and label those equations. We have, equation one is X, Y. Is equal to two. And then we have equation two is X squared plus Y squared is equal to five. Okay. Alright. Now we're told to use the substitution method and in the substitution method, what we wanna do, we wanna choose one variable? Okay, we're gonna isolate it in one of the equations and then substitute that into the other equation. So looking at our equations, if we want to isolate one of the variables equation one looks like it's going to be easier. We don't have to deal with any squares or square rooting. Okay. So let's go ahead and take equation one. So we have X. Y equals two. And let's isolate Y. So we're gonna have Y. Is equal to two over X. Okay, we're gonna call this equation one star so that we can refer to it. Alright, now we actually need to do our substitution method. So we've isolated why from equation one? We substitute that into equation two so that we find the solutions that satisfy both equations. Okay, so substitute one star into two. And what we find is we get X squared plus Y squared. So plus two over X squared is equal to five. Okay. All right let's give ourselves some more room to work. This gives us X squared plus four over X squared is equal to five. Okay, multiplying everything by X squared? We get X to the four plus four is equal to five X squared. Okay, so as long as X is not equal to zero, we are okay here. If X is equal to zero we're dividing by zero. That's gonna be a problem. And moving this over we have X to the four minus five, X squared plus four is equal to zero. Okay? And you might look at this and go this is a Equation with an exponent of four. I don't know how to deal with that. Well you do. Okay, it's okay. It's just like a quadratic equation if we change the variable. Okay, so let's let a equal to X squared. Okay. If A is equal to X squared X to the four is just gonna be a squared. So we can write our equation as a squared minus five, X squared is a plus four is equal to zero. Now we've rewritten our equation exponent for into just exponent two. We know how to deal with this. Okay, so let's factor we have a minus four times a minus one is equal to zero which gives us a values equal to one or four. Alright, so we have these a values equal to one or four. Now let's figure out what this means in terms of X. Ok. Alright, so are a values are equal to one or four? What does this mean in terms of A. Is equal to X squared? So this means we have x squared equals to one and we have x squared equal to four. Okay so we get X values of plus or minus one. Okay? We have to take the positive and negative square And X is equal to plus or -2. Okay. All right so it looked scary. We had an equation with an exponent of four but we were able to find our X values. Okay so now we have all of our X values. We need to find our corresponding y values and in order to do that we just need to substitute back into one of our equations. Okay we can choose either equation. We're gonna choose equation one star because we've already rearranged it and it just looks nicer and let's go back up to hear. Okay, give ourselves more room again and I'm just gonna rewrite the X values we found. So we found X equals plus or minus one or plus or minus two. Okay so I'm gonna leave those up there so that we can remember them. Alright so let's start with the first one. So we're gonna substitute our first x value one Into equation one star. Okay and again you can substitute it into either equation this one's gonna be easier to deal with so y is equal to two over X. Just 2/1 which is just two. Okay so this gives us the 20. two. Okay so we have the X. Value of one. Y value corresponding to We can do it again with x equals negative one. And we get Y is equal to 2/-1. She's gonna be negative too. So we have X. Value of negative one. Y. Value of negative two. So we get the point negative one comma negative two. Doing the same process again for our X value of two substituting into equation one star we have Y is equal to two over X which is 2 2/2 is equal to one. So we get the 10.2 comma one and one last time substituting X is equal to negative two into equation one star Y is equal to two over negative two. Let me give us a bit more room. Okay this gives us negative one and so we get the point negative two negative one. Okay so we found the y values first using the substitution method that we've plugged in to find the corresponding X values. Okay and now we have these four points and so our solution set can be written as these four points. We have minus two minus one minus one minus 212 And 2. 1. Okay so that is our solution. If you scroll back up to the question that is going to correspond with solution C. I hope this video helped. Thanks everyone for watching See you in the next one.

In Exercises 1–18, solve each system by the substitution method. xy=3, x^2+y^2=10

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