Question

give all solutions of the nonlinear system of equations, including those with nonreal complex components.

$4x^2 + y^2 = 8$

$4x^2 - y^2 = 0$

a. the solution set is \boxed{}.
( type an ordered pair. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. express complex numbers in terms of $i$. use a comma to separate answers as needed. )

b. there are infinitely many solutions.

c. the system is inconsistent. the solution set is $\varnothing$.

Explanation:

Step1: Add the two equations

We have the system:

$$\begin{cases} 4x^2 + y^2 = 8 \\ 4x^2 - y^2 = 0 \end{cases}$$

Adding the two equations to eliminate \(y^2\):
\((4x^2 + y^2)+(4x^2 - y^2)=8 + 0\)
Simplify: \(8x^2=8\)

Step2: Solve for \(x^2\)

Divide both sides of \(8x^2 = 8\) by 8:
\(x^2=\frac{8}{8}=1\)
Take square roots: \(x=\pm1\)

Step3: Solve for \(y^2\) using the second equation

From \(4x^2 - y^2 = 0\), we can express \(y^2 = 4x^2\)
When \(x = 1\): \(y^2=4(1)^2 = 4\), so \(y=\pm2\)
When \(x=- 1\): \(y^2=4(-1)^2 = 4\), so \(y = \pm2\)

Answer:

The solution set is \((1,2),(1, - 2),(-1,2),(-1,-2)\)