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.

$3 x^{2} + 5 y^{2} = 17$

$2 x^{2} - 3 y^{2} = 5$

Accepted Answer

Hello everybody. I hope you're doing all right today. So we're going to be looking at this math question that states provide all solutions for the given nonlinear system of equations including non-real complex components. And our system of equations is two X squared plus nine Y squared equals 38 and seven X squared minus three Y squared equals negative five. Now the other choices are ordered pairs and A is negative one comma negative one negative one, comma one, one comma negative one and one comma one B negative two comma negative one negative two comma one, two comma negative one and two comma one C negative one comma negative two negative one comma two, one common negative two and one comma two N D negative two comma negative two negative two comma 22 comma negative two and two comma two. Now, when working with ordered or a system equations, there's different ways that there are different things that we can do. In this case, I'm going to be using a method called elimination. So I'm going to set up both of my equations. One on top of the other with two X squared plus nine Y squared equals 38 on top and seven X squared minus three Y squared equals negative five on the bottom. And now imagine that you're adding both of these equations. And we have to think is there a number that we can use to multiply one of the equations that when adding the equations, one of the variables will be eliminated. And in this case there is, so we see that the top equation here has a nine Y squared. And we have in the bottom equation, a negative three Y squared, we know that we can get from 3 to 9 by multiplying by three or in this case, a negative three, since we have a negative three Y squared. So we can multiply the bottom equation by negative by or will multiply it by positive three because we want to eliminate the Y. So if we multiply it by negative three, we'll get positive Y and we have a or a positive nine and we have a positive nine on top. So we want to get a negative nine. So we'll multiply the bottom equation by positive three. And that will give us the top equation will remain the same. So we'll have two X squared plus nine Y squared equals 38 as our top equation. And then under it, all right, our bottom equation multiplied by three which is seven X squared, multiplied by three is 21 X squared. Then we had minus three Y squared, which is negative three Y squared multiplied by a positive three, which is minus nine Y squared. And that will be equal to negative five multiplied by three, which is negative 15. Now, when adding these two equations up, I have two X squared plus 21 X squared which is 23 X squared. And then we had a positive nine Y squared and now we have minus nine Y squared. So that will cancel out. And we have eliminated the Y variable. So finally, we'll have a 38 minus 15, which is 23. So we have 23 X squared equals 23 meaning X squared is equal to one and X is equal to the square root of one which is plus or minus one. So now we have our X values and we can plug them into one of our equations to figure out the Y values. So we'll do exactly that. So let's plug in positive one first and I'll be using the top equation that we were using. So two X squared plus nine Y squared equals 38. And first, I'll plug in positive one. So we have two multiplied by one squared plus nine Y squared equals 38. 1 squared would just be one and two multiplied by one is two. So we have two plus nine Y squared equals 38. Now, we can move the two over from the left hand side to the right hand side and we'll have nine Y squared equals 36. And if we divide both sides by nine, we have Y squared equals four or Y equals the square root of four, which is equal to plus or minus two. So now we can test our second X, which in this case was negative one. So once again, we have two X squared plus nine Y squared equals 38. And we plug in negative one for the X. So we have two multiplied by negative one. In this case, squared plus nine Y squared equals 38 and negative one squared will be positive one because a negative multiplied by a negative is a positive. So we have positive one multiplied by two is two. We have two plus nine Y squared equals 38. And this matches one of the lines that we had in our previous equations. So we know that Y will once again be equal to plus or minus two. So our ordered pairs will be when X is one will have a positive tool or we can also have an X of one and a negative two. And we can have for the other order repairs. When we have an X of negative one, we can have a Y of positive two or an X of negative one and A Y of negative two. So our ordered pairs will be one, comma 21, comma negative two negative one, comma two and negative one. Comma negative two. And now we can just um compare those with our answer choices. And when we do that, we can see that answer choice C matches our order pairs and will be the answer to this question. 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.
$3 x^{2} + 5 y^{2} = 17$
$2 x^{2} - 3 y^{2} = 5$

Key Concepts

Nonlinear Systems of Equations

Substitution and Elimination Methods

Complex Number Solutions

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