A reflection symmetry can save you from solving every point in a nonlinear system. Change to in each equation. If both equations stay true, every solution has a partner with the opposite -coordinate. Pair those coordinates before adding them; keeping only the points with positive leaves out valid solutions.
Hints
- Hint 1
A reflection across the -axis changes to without changing its height. Does squaring give a different result when you change its sign?
- Hint 2
Absolute value measures distance from , so and are equal. Check whether the second equation still holds when changes to .
- Hint 3
If both equations stay true, each solution with a nonzero -coordinate has a mirror partner. Add the two -coordinates as a pair rather than finding every point. What would a coordinate of contribute?
Step-by-step
Pair solutions by symmetry
Step 1Reflect a point on the circle
No Desmos needed. Reflection symmetry gives the sum without finding every point. A reflection across the -axis changes to . Squaring removes the sign, so if is on the circle, its reflection is too:
- Step 2
Check the other equation
Absolute value is distance from , so . If satisfies the second equation, then
Both equations stay true for and . So each reflected point is a solution to the whole system, not only to the circle.
- Step 3
Add the paired coordinates
Add each pair's -coordinates:
Every nonzero -coordinate cancels with its opposite; a coordinate of would add nothing. So the sum of all possible -coordinates is . Grid in 0.