Use technology to graph the system, then answer the question.

Question

Using a graph, how many solutions are there in this linear-quadratic system of equations?

y=−x2−6x+2

y=−111x+12

(1 point)
Responses

The graphs intersect in one point, so there is 1 solution.
The graphs intersect in one point, so there is 1 solution.

The graphs never intersect, so there is no solution.
The graphs never intersect, so there is no solution.

The line crosses the parabola in two locations, so there are 2 solutions.
The line crosses the parabola in two locations, so there are 2 solutions.

The parabola and line are the same graph so there are many solution points.

Answer 1

To determine how many solutions exist in the linear-quadratic system of equations given by

1. \( y = -x^2 - 6x + 2 \) (the parabola)
2. \( y = -111x + 12 \) (the line)

you can graph both equations and look for points of intersection.

1. The first equation represents a downward-facing parabola. The shape of the parabola will open downwards due to the negative coefficient of \( x^2 \).
2. The second equation is a straight line with a negative slope.

Now, when you graph these equations:

• The parabola will curve downwards.
• The line will cross through the parabola.

You will analyze if the line intersects the parabola. The solutions of the system correspond to the points where the line intersects the parabola.

Through graphing it, you might find that:

1. The line intersects the parabola twice. This would mean there are 2 points where the two equations satisfy each other, yielding 2 solutions.
2. If the line were to cross the parabola at one point, it would mean there is exactly one solution (this might occur at the vertex of the parabola).
3. If the line is entirely above or below the parabola, there would be no points of intersection, resulting in no solution.

Therefore, depending on your graph:

If the line crosses the parabola in two locations, the correct answer is: "The line crosses the parabola in two locations, so there are 2 solutions."

If you have access to graphing software or a graphing calculator, you can visualize the exact intersections and confirm this.