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Question 7 of 30

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M:C:MC:7

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SAT

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The figure presents the graph of a circle, a parabola, and a line in the xy plane. The horizontal axis is labeled x, the vertical axis is labeled y, and the origin is labeled O. The integers negative 3 through 3 appear on both axes. The circle has its center at the origin and radius of about two point two. The parabola has its vertex on the y-axis at negative 3 and opens up. The circle and parabola intersect at four points, of which two are below the x-axis and two are above the x-axis. Of the two points of intersection below the x-axis, one is to the left of the y-axis and one is to the right of the y-axis. Of the two points of intersection above the x-axis, one is to the left of the y-axis and one is to the right of the y-axis. The line slants upward and to the right, and passes through two of the four points of intersection where the circle and parabola meet, one below the x-axis and to the left of the y-axis, and one above the x-axis to the right of the y-axis. In other words, the three graphs intersect at two points.

 

table row cell x squared end cell plus cell y squared end cell equals 5 row y equals cell x squared end cell minus 3 row x minus y equals 1 end table

A system of three equations and their graphs in the xy-plane are shown above. How many solutions does the system have?

Select an Answer

One

Correct Answer: 
No

Two

Correct Answer: 
Yes

Three

Correct Answer: 
No

Four

Correct Answer: 
No

Choice B is correct. The solutions to the system of equations are the points where the circle, parabola, and line all intersect. These points are (-1,-2) and (2,1), and these are the only solutions to the system.

Choice A is not the correct answer. This answer may reflect the misconception that a system of equations can have only one solution.

Choice C is not the correct answer. This answer may reflect the misconception that a system of equations has as many solutions as the number of equations in the system.

Choice D is not the correct answer. This answer may reflect the misconception that the solutions of the system are represented by the points where any two of the curves intersect, rather than the correct concept that the solutions are represented only by the points where all three curves intersect.

Question Difficulty: 
easy
Objective: 

Students must demonstrate their conceptual understanding of graphical representations of variable relationships.