What is the geometric interpretation of the system of equations y=y^2-x+1 y=3-2x

The geometric interpretation of the system means to check
if the given line and parabola are intercepting in 2 points, one point or any point at
all.

We'll solve the system putting the 1st equation equal
to the 2nd one.

x^2 - x + 1 = 3 -
2x

We'll move all terms to the
left:

x^2 - x + 1 - 3 + 2x =
0

We'll combine like
terms:

x^2 + x - 2 = 0

We'll
apply quadratic formula:

x1 = [-1+sqrt(1 +
8)]/2

x1 = (-1+3)/2

x1 =
1

x2 = -2

The y va;ues
are:

y1 = 3 - 2x

y1 =  3 -
2

y1 = 1

y2 = 3 +
4

y2 = 7

The solutions of the
system are the pairs: (1 , 1) and (-2 ,
7).

Since the system has 2 solutions, that
means that there are 2 intercepting points of the line and parabola and they are:  (1 ,
1) and (-2 , 7).