Question

question 22 of 25
what are the solutions to the nonlinear system of equations below?
check all that apply.
$x^2 + y^2 = 9$
$\frac{x^2}{3^2} - \frac{y^2}{3^2} = 1$

a. $(-3, -3)$
b. $(0, 3)$
c. $(3, 3)$
d. $(-3, 0)$
e. $(0, -3)$
f. $(3, 0)$

Explanation:

Step1: Simplify the second equation

First, simplify the hyperbola equation:
$$\frac{x^2}{3^2} - \frac{y^2}{3^2} = 1 \implies \frac{x^2 - y^2}{9} = 1 \implies x^2 - y^2 = 9$$

Step2: List the system of equations

We now have the system:

1. $x^2 + y^2 = 9$
2. $x^2 - y^2 = 9$

Step3: Add the two equations

Add to eliminate $y^2$:
$$(x^2 + y^2) + (x^2 - y^2) = 9 + 9$$
$$2x^2 = 18$$
$$x^2 = 9 \implies x = 3 \text{ or } x = -3$$

Step4: Solve for $y$

Substitute $x^2=9$ into the first equation:
$$9 + y^2 = 9$$
$$y^2 = 0 \implies y = 0$$

Step5: Verify solutions

The solutions are $(3,0)$ and $(-3,0)$. Check these against the options:

• For $(-3,0)$: $(-3)^2 + 0^2 = 9$, $\frac{(-3)^2}{9} - \frac{0^2}{9} = 1$, which satisfies both equations.
• For $(3,0)$: $3^2 + 0^2 = 9$, $\frac{3^2}{9} - \frac{0^2}{9} = 1$, which satisfies both equations.

Answer:

D. (-3, 0)
F. (3, 0)