Solve each nonlinear system of equations. Give all solutions, inc...

Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
2xy + 1 = 0 
x + 16y = 2

Accepted Answer

Hello everybody. We're doing all right today today we're going to be looking at this question that states provide all solutions for the given nonlinear system of equations including non-real complex components. Where our equations are, we'll label number equation number one as five X Y plus one equals zero. And equation number two will be X plus 40 Y equal to. Now the answers provided are all sets of two X and Y components with letter A being negative four comma one divided by 10 and two comma negative one divided by 20 B is negative two comma one divided by and four comma negative one divided by 20 C is one divided by 10 comma negative two negative one divided by comma four and D is one divided by 10 comma negative four and negative one divided by 20 comma two. Now to software these variables or these components, we need to start playing with the equations that we have. So first, I'm going to do some uh manipulation with equation number one. So we had that five X Y plus one equals zero. If we bring the one over from the left hand side to the right hand side by subtracting, we'll have that five, Y equals negative one. And if we divide both sides by five, now we'll have that X Y equals negative one, divided by five or negative 1/5. And now if we divide both sides by X, we'll have that Y is equal to negative one, divided by five X. So now we have an equation for Y and we can plug that into our equation number two. So we'll have the X plus 40. And first, this is where the Y would come in. So now we'll have 40 multiplying the negative one divided by five X and that'll still be equal to two. And if we distribute 40 multiplying a negative one divided by five X, it's the same as saying 40 divided by negative five X, which will be so we'll have now X plus or minus, since we're dividing by a negative number, we'll have X minus eight divided by X equals two. And I will multiply everything by X to get rid of the X and the denominator of our fraction. So we'll have X squared minus eight equals two X. And we'll move everything to one side of our equation. So we move over the two X from the right hand side to the left hand side to have X squared minus two, X minus eight equals zero. And now I can do some factoring where I'll have two parentheses and inside of both parentheses, the first term will be an X because I know they have to multiply it to get the X squared that we have. And now we have to think of two numbers that when multiplied by each other will give me negative eight, which is the last term in our equation, but would add up to negative two, which is the middle term in our equation. And those two numbers will be negative four which will go into into the first parentheses and positive two which will go into the second parentheses. So in the first parentheses, we have X minus four and that parentheses will be multiplying another parentheses that has X plus two inside of it. And like we said, so negative four multiplied by positive two will be negative eight, which is the last term that we need and negative four plus a po or yeah, negative four plus two will be negative two, which is the middle term that we need. So now we can set each of those parentheses equal to zero. So we'll have X minus four equals zero and X plus two equals zero to get that X can be positive for or X equals negative two. So those would be two possible answers for a question. And now we can plug those axes in with the Y values would be for each one. So if X equals negative two, Y equals Y equals negative one divided by five, multiplying a negative two. And there we just used our Y equals negative one divided by five X equation where we just plugged in negative two for the X, which means that, that will be equal to negative one divided by negative 10, which means that Y is equal to one divided by or negative two comma one divided by 10, our first set. And now we plug in if X equals four, we have that Y is equal to negative one divided by five, multiplied by four, which will be negative one divided by 20. So we'll have that Y equals negative one divided by 20 or the point four comma negative one divided by 20. And that will be our second set for our equation. So now we just compare those two, our answer choices and we have answer choice B matching with our answers. A set of negative two comma one divided by 10 and a set of four comma negative one divided by 20. So I hope that was helpful. And until next time.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. 2xy + 1 = 0 x + 16y = 2

Key Concepts

Nonlinear Equations

Complex Numbers

Graphical Interpretation

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