To solve the system of equations:
[tex]\[
\left\{
\begin{array}{l}
x^2 + y^2 = 16 \\
y + 4 = x^2
\end{array}
\right.
\][/tex]
follow these steps:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] using the second equation:
[tex]\[
y + 4 = x^2 \quad \Rightarrow \quad y = x^2 - 4
\][/tex]
2. Substitute [tex]\( y \)[/tex] into the first equation:
[tex]\[
x^2 + y^2 = 16 \quad \Rightarrow \quad x^2 + (x^2 - 4)^2 = 16
\][/tex]
3. Simplify and solve the equation:
[tex]\[
x^2 + (x^2 - 4)^2 = 16
\][/tex]
Expand and simplify:
[tex]\[
x^2 + (x^2 - 4)^2 = 16 \\
x^2 + (x^4 - 8x^2 + 16) = 16 \\
x^4 - 7x^2 + 16 = 16 \\
x^4 - 7x^2 = 0
\][/tex]
Factor the equation:
[tex]\[
x^2(x^2 - 7) = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 = 0 \quad \text{or} \quad x^2 = 7 \\
x = 0 \quad \text{or} \quad x = \pm \sqrt{7}
\][/tex]
4. Find the corresponding [tex]\( y \)[/tex] values for each [tex]\( x \)[/tex] solution:
Using [tex]\( y = x^2 - 4 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0^2 - 4 = -4
\][/tex]
- For [tex]\( x = \sqrt{7} \)[/tex]:
[tex]\[
y = (\sqrt{7})^2 - 4 = 7 - 4 = 3
\][/tex]
- For [tex]\( x = -\sqrt{7} \)[/tex]:
[tex]\[
y = (-\sqrt{7})^2 - 4 = 7 - 4 = 3
\][/tex]
5. Summarize the solutions as coordinate pairs:
[tex]\[
(x, y) = (0, -4), (\sqrt{7}, 3), (-\sqrt{7}, 3)
\][/tex]
In conclusion, the solutions to the system of equations are:
[tex]\[
(0, -4), (\sqrt{7}, 3), (-\sqrt{7}, 3)
\][/tex]
