A solution to a system is a point that satisfies both equations, so it appears where their graphs intersect. For a line and a cubic curve, graph both equations in Desmos and count every crossing. A cubic can bend twice; don't treat it like a parabola and assume there can be at most two intersections.
Hints
- Hint 1
A solution is a pair that makes both equations true. On a graph, the point must sit on both the curve and the line. Where do you see points like that?
- Hint 2
A curve with an term can bend twice. Scan across the whole graph, including beside the -axis, so you don't miss a place where the line crosses it.
Step-by-step
Count the intersections in Desmos
Step 1Graph the cubic curve
Type in Desmos. This cubic has an term, and its graph rises, falls, then rises again. Desmos shows two turns, so don't assume a line can cross it at most twice.
- Step 2
Add the line and count shared points
Add , then click each place the graphs cross. Desmos shows points near , , and . An intersection lies on both graphs, so each shared point is one solution. Count the crossing near the -axis too: the system has exactly solutions. Choice D.