Question

Solve the following nonlinear system algebraically. be sure to check for non-real solutions.

Answer 1

`\begin{gathered} \text{Given} \\ x^2+y^2=8\text{ \lparen first equation\rparen} \\ 2x^2+4y^2=34\text{ \lparen second equation\rparen} \end{gathered}`

Use substitution method to and solve in terms of x using the first equation

`\begin{gathered} x^2+y^2=8 \\ x^2=8-y^2 \end{gathered}`

Next, substitute it to the second equation

`\begin{gathered} 2x^2+4y^2=34 \\ 2(8-y^2)+4y^2=34 \\ 16-2y^2+4y^2=34 \\ 2y^2=34-16 \\ 2y^2=18 \\ (2y^2)/(2)=(18)/(2) \\ y^2=9 \end{gathered}`

Then, substitute it back to first equation and solve for x

`\begin{gathered} x^2+y^2=8 \\ x^2+9=8 \\ x^2=8-9 \\ x^2=-1 \end{gathered}`

Now we have the following solutions

`x^2=-1\text{ and }y^2=9`

Get the square root of both x's and y's to get the solution

Getting the combination of ordered pairs we have the following solutions

`\begin{gathered} (i,3) \\ (i,-3) \\ (-i,3) \\ (-i,-3) \end{gathered}`