A nonlinear system pairs a curve with another equation, and each intersection is a point that satisfies both. Type both equations into Desmos and click every intersection before adding the coordinates the question names. A circle is symmetric about its center, but a line that misses the center need not meet it at points with opposite -coordinates.
Hints
- Hint 1
An intersection lies on both graphs, so it solves both equations. A line can cut a circle twice. How many meeting points do you need to check when the question asks about all solutions?
- Hint 2
An ordered pair lists first and second. Once you click each intersection, take the second number from each point before you add.
Step-by-step
Approach 1: Graph both equations and add the y-coordinates
Step 1Find every intersection
Type and on separate Desmos lines. Click both intersections. Desmos shows and . Each point is a solution because it lies on both graphs.
- Step 2
Add the requested coordinates
In an ordered pair , is the second number. Take both -coordinates, and , rather than stopping at one intersection:
For a coordinate sum, take the requested coordinate from every intersection. The sum of the -coordinates is . Choice C.
Approach 2: Use the sum of a quadratic’s roots
Step 1Express x in terms of y
Subtract from to express in terms of :
- Step 2
Make an equation in y alone
Now substitute, or replace with , in the circle equation:
- Step 3
Expand the square
The square means . Multiply it out:
- Step 4
Combine like terms
The two terms are like terms, so add them:
- Step 5
Move the constant to one side
Subtract from both sides:
- Step 6
Reduce the quadratic
Divide every term by :
- Step 7
Read the sum from the coefficients
The roots of this quadratic are the -values that make it zero. For , their sum is : expanding gives a middle coefficient of . Here and , so no individual roots are needed:
The sum of the solutions’ -coordinates is . Choice C.