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.

$x^{2} + y^{2} = 02$

$x^{2} - 3 y^{2} = 0$

Accepted Answer

Hey, everyone. Here we are asked to provide all solutions for the given nonlinear system of equations including non-real complex components. Here. The first given equation is two X squared plus seven Y squared is equal to zero. And the second given equation is eight X squared minus three Y squared is equal to zero. So here we have four answer choice options. Each including the solution set of an ordered pair. Answer choice A is the solution set of the ordered pair, one and seven halves. Answer B is the square root of three divided by two and one answer C is zero and one and answer D is zero and zero. So here we can utilize the substitution method to solve, solve either of our given equations for either X squared or Y squared. So we can, here we can select the second equation and solve for X squared. So to solve for X squared, we simply add three Y squared to both sides of the equation. So we have A X squared is equal to three Y square squared. Now continuing to solve for X squared, we divide both sides by the coefficient of X squared which is eight. So we are left with X squared is equal to 3/8 times Y squared. So now with this expression, we found four X squared, we need to take it and substitute it into our first equation. And then we can solve four Y. So our first equation, we had two X squared plus seven Y squared is equal to zero. And now substituting in our expression, we have two times the quantity of 3/8 Y squared plus seven Y squared is equal to zero. Now, simplifying our product, we have 3/4 Y squared plus seven Y squared is equal to zero, combining like terms. We have 31 4th Y squared is equal to zero, dividing both sides by the coefficient which is 31 4th, we are left with Y squared is equal to zero. And now, since Y squared is equal to zero, if we were to take the square root of both sides, we can clearly see that Y is just equal to zero. So here we have found our value for Y and with this value, we can now substitute it back into our expression we found for X squared. So before we had X squared is equal to 3/8 times Y squared. Well, now we have X squared is equal to 3/8 times zero. And now just simplifying our product, we have X squared is equal to zero and taking the square root of both sides, we clearly see that X is just equal to zero. So now we have found our value for X and our value for Y. So now we can rewrite our solution as the solution set of the ordered pair zero and zero. So with this solution, we are left with answer choice D where again, we have the solution set of the ordered pair zero and zero. 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.
$x^{2} + y^{2} = 02$
$x^{2} - 3 y^{2} = 0$

Key Concepts

Nonlinear Systems of Equations

Complex Numbers and Imaginary Solutions

Substitution and Elimination Methods

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