Question

Consider the following system of equations:
10 + y = 5x + x2
5x + y = 1
The first equation is an equation of a
✔ parabola
.

The second equation is an equation of a
✔ line
.

How many possible numbers of solutions are there to the system of equations?
0
1
2
3
4
infinite

What are the solutions to the system? (
, -4) and (
,
)

Answer 1

A parabola and a line can meet in 0, 1, or 2 points (they cannot meet in 3 or more unless the "parabola" degenerates to a line, which it doesn't here).

Solve: from 5x + y = 1 → y = 1 − 5x. Substitute into 10 + y = 5x + x^2:
10 + (1 − 5x) = 5x + x^2 → 11 − 5x = 5x + x^2 → x^2 + 10x − 11 = 0.
x = [−10 ± √144]/2 = (−10 ± 12)/2 → x = 1 or x = −11.

Then y = 1 − 5x gives y = −4 when x = 1, and y = 56 when x = −11.

Solutions: (1, −4) and (−11, 56).