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.

$5 x^{2} - 2 y^{2} = 25$

$10 x^{2} + y^{2} = 50$

Accepted Answer

Hello everybody I I today that are going to look at this math question that states provide all solutionss for the given nonlinear nonlinear system of equations including non-real complex components. Now, the system of equations is nine X squared minus two Y squared equals 12 X squared plus Y squared equals 84. And the answer choice provided are sets of ordered pairs with a being negative square root of seven comma zero and positive scored of seven comma zero B zero comma negative squared of seven N zero, comma positive squared of seven C negative squared of eight comma zero and positive squared of eight comma zero and D zero comma negative squared of eight N zero comma square root of eight. Now we're looking at this question, I see that we have two system of equations. So there's different way or or two equations here that we can approach in different ways. In this case, I'm going to be using the elimination method. So to use the elimination method, we need to see if there is a number that we can use to multiply one of the equations that when multiplied by that number will have help us cancel out that variable. So in this case, I'm going to set up our equations one on top of the other. So on top I have nine X squared minus two Y squared equals 63. And under it, I have 12 X squared plus Y squared equals 84. So I have it like this. So we can imagine that we're adding both equations. Now, is there a number that I can use to multiply one of the equations that when adding both equations together it, one of the variables were canceled. Well, there is I can multiply the bottom equation by two. So let's do that. So if we do that, we'll have, we'll rewrite this as nine X squared minus two Y squared equals 63. That top equation will remain the same. And now we multiply the bottom equation by two, we'll have two X squared multiplied by two will be 24th X squared and Y squared multiply Y two will be plus two Y squared and then we'll have equal to 84 multiplied by two, which is 100 and 68. And now if we add both of these equations up, we have nine X squared plus 24 X squared, which is 33 X squared, negative two Y squared plus two Y squared is zero. So we have canceled out the Y and then 63 plus 100 and 68 will be 231. So now we have that 33 X squared equals 231. So we can solve for X here. If we divide both sides by 33 we have that X squared equals seven. And then we take the square root of both sides. So we have that X is equal to plus or minus the square of seven. So all we have to do now is try uh test the X in one of our equations and solve for the Y. So that's exactly what we'll do. We have nine X squared minus uh two Y squared equals 63. That's the equation that I'll be using. So we have nine and we'll try positive square root of seven first. So we'll have nine multiplied by the positive square root of seven squared minus two Y squared equals 63. The square root of seven squared is seven. So we'll have nine multiplied by seven minus two Y squared equals 63. Nine, multiplied by seven is 63. So I have 63 minus two Y squared equals 63. And then we'll have, we can bring the 63 over from the left hand side of the equation to the right hand side. So we'll have that negative two Y squared equals zero, which means that Y must equal zero. So we have our first order pair of X of positive squared of seven comma zero. Now, we can try our second X So we have nine multiplying the negative squared of seven squared minus to A Y squared equals 63. And now same concept, we have the negative squared of seven squared would just be seven positive seven because a negative multiplied by a negative is a positive. And then we'll have minus two Y squared equals 63. And we see that this line is the same as a line we had in the previous um steps. So we know that this will lead to Y equals zero as well or an ordered pair of negative squared of seven comma zero. So we have our two ordered pairs, negative set, negative square root of seven comma zero and positive seven comma zero, which in this case is answer choice A. 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.
$5 x^{2} - 2 y^{2} = 25$
$10 x^{2} + y^{2} = 50$

Key Concepts

Nonlinear Systems of Equations

Substitution and Elimination Methods

Complex Solutions in Algebra

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