A system asks for points that satisfy two equations at once. When a line and a parabola intersect at two points, graph both equations in Desmos and click each crossing. If the question asks for the sum of the -coordinates, add the first number from each point. Don’t use the parabola’s -intercepts; those may not lie on the line.
Hints
- Hint 1
An intersection is a point on both graphs, so its coordinates make both equations true. Graph the line and the parabola together. Where do they cross?
- Hint 2
In an ordered pair , is the first coordinate. Click both intersections, then pick the first number from each point before you add.
Step-by-step
Approach 1: Read both intersections in Desmos
Step 1Graph the line and the parabola
Type and on separate lines. Desmos shows a parabola and a line crossing twice. You need their shared points, not the places where the parabola meets the -axis.
- Step 2
Read the two shared points
Click both intersections. Desmos labels them approximately and . The first number in each pair is its -coordinate, so the two numbers you need are about and .
- Step 3
Add the x-coordinates
Add the first coordinates from the two point labels:
Desmos rounds point labels, so this calculation gives an estimate; the coefficient method below confirms the exact sum. The sum of the intersection -coordinates is . Choice C.
Approach 2: Get the exact sum from the coefficients
Step 1Set the two y-values equal
At an intersection, both equations give the same . So the two expressions on the right must be equal:
- Step 2
Put the quadratic in standard form
Subtract from both sides:
Add to both sides:
Its solutions are the intersection -coordinates, because they make the two original -values equal.
- Step 3
Use the sum of the roots
The roots of a quadratic are the -values that make it equal . For , their sum is : expanding gives the middle coefficient . Here and , so substitute:
The two intersection -coordinates sum to . Choice C.