A line and a parabola form a nonlinear system: they can meet at more than one point. A solution must lie on both graphs, so graph the two given equations in Desmos and click their intersections. Then match the ordered pair, with first and second. A point on the line alone may not satisfy the parabola.
Hints
- Hint 1
A solution to a system is one ordered pair that makes both equations true. Since each equation gives a value for , what must be true of those two values at a solution?
- Hint 2
In Desmos, an intersection is a point shared by two graphs. Graph both equations and click every crossing. Which displayed ordered pair appears among the choices?
Step-by-step
Approach 1: Graph the intersections
Step 1Identify what a solution must satisfy
Both equations say what equals, so at a solution their right sides must agree:
Matching one equation isn't enough; the ordered pair must work in both.
- Step 2
Read and match the shared points
Type and on separate lines in Desmos. Click both intersections: Desmos shows and . These are the pairs that satisfy both equations, and only appears in the choices. Choice D.
Approach 2: Factor to explain the two crossings
Step 1Collect the terms
No Desmos needed. This quadratic factors by hand. Start with the equality between the two expressions for . Subtract from both sides:
Add to both sides:
- Step 2
Write the quadratic as a product
To factor is to rewrite an expression as a product. The numbers and multiply to and add to , so factor the quadratic:
- Step 3
Find both x-coordinates
By the zero-product rule, a product is zero when at least one factor is zero:
Solve each equation, keeping both values:
- Step 4
Find y and choose the listed pair
Use the simpler equation, , to find the matching y-coordinate for each value of :
The solutions are and , and is the one listed. Choice D.