We are tasked with solving the system of nonlinear equations:
\[
\begin{aligned}
1. & \quad x^2 + y^2 = 20 \\
2. & \quad x^2 - y^2 = -2
\end{aligned}
\]
Step 1: Add the two equations
Adding the two equations eliminates \( y^2 \):
\[
(x^2 + y^2) + (x^2 - y^2) = 20 + (-2)
\]
Simplify:
\[
x^2 + x^2 + y^2 - y^2 = 18
\]
\[
2x^2 = 18
\]
Solve for \( x^2 \):
\[
x^2 = \frac{18}{2} = 9
\]
Thus:
\[
x^2 = 9 \implies x = \pm 3
\]
Step 2: Subtract the second equation from the first
Subtracting the second equation from the first eliminates \( x^2 \):
\[
(x^2 + y^2) - (x^2 - y^2) = 20 - (-2)
\]
Simplify:
\[
x^2 + y^2 - x^2 + y^2 = 22
\]
\[
2y^2 = 22
\]
Solve for \( y^2 \):
\[
y^2 = \frac{22}{2} = 11
\]
Thus:
\[
y^2 = 11 \implies y = \pm \sqrt{11}
\]
Step 3: Combine the solutions
From \( x^2 = 9 \), we have \( x = \pm 3 \).
From \( y^2 = 11 \), we have \( y = \pm \sqrt{11} \).
The possible combinations of \( (x, y) \) are:
\[
(x, y) = (3, \sqrt{11}), (3, -\sqrt{11}), (-3, \sqrt{11}), (-3, -\sqrt{11})
\]
Step 4: Verify the solutions
We substitute each pair back into the original equations to ensure they satisfy both.
1. For \( (3, \sqrt{11}) \):
\[
x^2 + y^2 = 3^2 + (\sqrt{11})^2 = 9 + 11 = 20 \quad \text{(satisfied)}
\]
\[
x^2 - y^2 = 3^2 - (\sqrt{11})^2 = 9 - 11 = -2 \quad \text{(satisfied)}
\]
2. For \( (3, -\sqrt{11}) \):
\[
x^2 + y^2 = 3^2 + (-\sqrt{11})^2 = 9 + 11 = 20 \quad \text{(satisfied)}
\]
\[
x^2 - y^2 = 3^2 - (-\sqrt{11})^2 = 9 - 11 = -2 \quad \text{(satisfied)}
\]
3. For \( (-3, \sqrt{11}) \):
\[
x^2 + y^2 = (-3)^2 + (\sqrt{11})^2 = 9 + 11 = 20 \quad \text{(satisfied)}
\]
\[
x^2 - y^2 = (-3)^2 - (\sqrt{11})^2 = 9 - 11 = -2 \quad \text{(satisfied)}
\]
4. For \( (-3, -\sqrt{11}) \):
\[
x^2 + y^2 = (-3)^2 + (-\sqrt{11})^2 = 9 + 11 = 20 \quad \text{(satisfied)}
\]
\[
x^2 - y^2 = (-3)^2 - (-\sqrt{11})^2 = 9 - 11 = -2 \quad \text{(satisfied)}
\]
All four pairs satisfy both equations.
Final Answer:
\[
\boxed{(3, \sqrt{11}), (3, -\sqrt{11}), (-3, \sqrt{11}), (-3, -\sqrt{11})}
\]
Parent Tip: Review the logic above to help your child master the concept of solving nonlinear systems worksheet.
