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

Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

$x^{2} + y^{2} = 102$

$x^{2} - y^{2} = 17$

Accepted Answer

Hey, everyone here we are to provide all solutions for the given nonlinear system of equations including non-real complex components. Here. The first given equation is X squared plus Y squared is equal to 65. And the second given equation is three X squared minus Y squared is equal to negative one. Here we have four answer choice options. Each of them being a solution set including four different ordered pairs. So here looking at our two given equations, we see that both of our Y terms are the same aside from their sign. So here we can utilize the elimination method to solve four X by eliminating our Y terms. So eliminating our Y terms, since they have opposite signs, we just need to add our two equations together. And now going term by term, we have X squared plus three X squared which gives us four X squared, we then have Y squared minus Y squared, which gives us zero as we intended. And then on the right side, we have 65 plus negative one which gives us 64. So now we have a third equation after adding our two equations together where we have four, X squared is equal to 64. Now just solving four X, we first divide both sides by four and we have X squared is equal to 16. Now, continuing to solve for X, we take the square root of both sides to get rid of our square exponent. And that means we need to add a plus or minus in front of the square root on the right hand side. So we have X is equal to plus or minus the square root of 16. And this simplifies to just plus or minus four. So here we have two different possible solutions for X and so we need to input both possible solutions into our first equation to solve for all of the possible values of Y. So starting when X is equal to four, again, our original equation, we have X squared plus Y squared is equal to 65. And now just inputting this value of X, we have four squared plus Y squared is equal to 65. Now just simplifying the square, we have 16 plus Y squared is equal to 65. Now, just solving for Y, we first subtract 16 from both sides and we have Y squared is equal to 49. Now solving for Y, we need to take the square root of both sides. And since we are adding the square root on the right side of the equation, we need to have a plus or minus before the square root. So we have Y is equal to plus or minus the square root of 49 which simplifies two plus or minus seven. And so now we see that when X is equal to four, Y is equal to either positive seven or negative seven. So now we want to move on to input in our other value of X, which is when X is equal to negative four. And again, using the same equation, we now have the quantity, the quantity of negative four squared plus Y squared is equal to 65. Now simplifying the squared term, we again have 16 plus Y squared is equal to 65. And now we see we have the same solution as previously. So we find that Y again is equal to plus or minus seven. So here when X is equal to negative four, Y is equal to either positive seven or negative seven. So now with all of these possible solutions, we can summarize our solutions as a solution set containing four different ordered pairs. So starting when X is positive four, we have positive four and seven as one ordered pair. We also have four and negative seven. And now when X is equal to negative four, we have negative four and positive seven and we also have negative four and negative seven. So with this final solution set, we are left with answer choice B where again, we have the solution set of four ordered pairs, one of them is negative four and negative seven and then negative four and seven, then four and negative seven and lastly four and seven. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$x^{2} + y^{2} = 102$
$x^{2} - y^{2} = 17$

Key Concepts

Nonlinear Systems of Equations

Substitution and Elimination Methods

Complex Number Solutions

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