Question

use the graph to answer the question. what are the solutions to this system of equations? a. (-3, 0) and (2, 5) b. (-3, 0) and (3, 0) only c. (-3, 0), (3, 0), and (0, 9) d. (-3, 0), (3, 0), (0, 3), and (0, 9)

Explanation:

Step1: Recall system solution definition

Solutions to a system of equations are the intersection points of their graphs.

Step2: Identify intersection points

- The parabola (curved graph) intersects the x - axis at \((- 3,0)\) and \((3,0)\)? Wait, no, looking at the graph, the parabola has a vertex at \((0,9)\) (since it's symmetric about the y - axis and the highest point is at \(x = 0,y=9\)) and intersects the x - axis? Wait, no, the straight lines and the parabola: Wait, the system consists of a parabola and a line (or two lines? Wait, the graph has a parabola and two lines? Wait, no, the problem is a system of equations, so the solutions are the points where the graphs intersect. Looking at the options, let's analyze each:
- Option A: \((-3,0)\) and \((2,5)\): \((2,5)\) is not an intersection point of all graphs? Wait, no, the system is likely the parabola and the line. Wait, the parabola \(y=-x^{2}+9\) (since vertex at \((0,9)\) and opens downward) and the line: let's find the intersection. The line passes through \((-3,0)\) and \((0, - 3)\)? Wait, no, looking at the graph, the intersection points of the parabola (curved) and the straight line (the one with negative slope) and maybe another line? Wait, no, the solutions to the system are the points where all the graphs (in the system) intersect. Wait, the options: Option C is \((-3,0)\), \((3,0)\), and \((0,9)\)? No, wait, \((0,9)\) is the vertex of the parabola. Wait, no, let's re - examine: The parabola \(y=-x^{2}+9\) (since when \(x = 0,y = 9\), \(x=\pm3,y = 0\)) and the line: let's see the line passes through \((-3,0)\) and \((0,-3)\)? No, wait, the other line passes through \((-3,0)\) and \((0,3)\)? Wait, no, the key is that the solutions to the system are the intersection points of the graphs. The parabola \(y=-x^{2}+9\) intersects the x - axis at \((-3,0)\) and \((3,0)\), and intersects the line (or lines) at? Wait, no, the options: Option C has \((-3,0)\), \((3,0)\), and \((0,9)\)? Wait, no, \((0,9)\) is the vertex of the parabola. Wait, maybe the system is the parabola and two lines? No, the problem says "this system of equations", so the graphs involved are the parabola and the line (or two lines). Wait, looking at the options, Option C: \((-3,0)\), \((3,0)\), and \((0,9)\). Wait, \((0,9)\) is the vertex of the parabola, and \((-3,0)\) and \((3,0)\) are the x - intercepts of the parabola, and maybe the y - intercept of the parabola is part of the system? No, the solutions to the system are the intersection points. Wait, maybe the system is the parabola \(y=-x^{2}+9\) and the two lines? No, the correct approach: Solutions to a system of equations are the points where the graphs intersect. The parabola \(y =-x^{2}+9\) intersects the x - axis at \((-3,0)\) and \((3,0)\), and intersects the y - axis at \((0,9)\). Also, the line (or lines) in the system: if the system includes the parabola and the two lines, but the options show that Option C is \((-3,0)\), \((3,0)\), and \((0,9)\). Wait, maybe the system is the parabola and the two axes? No, the problem is a system of equations (probably a quadratic and a linear, or two linears and a quadratic). But from the options, the correct solutions are the intersection points of all the graphs in the system. The parabola \(y=-x^{2}+9\) has x - intercepts at \((-3,0)\) and \((3,0)\) and y - intercept at \((0,9)\), and if the system includes the lines that pass through these points, then the solutions are \((-3,0)\), \((3,0)\), and \((0,9)\), which is Option C.

Step3: Eliminate other options

- Option A: \((2,5)\) is not an intersection point.
- Option B: Only \((-3,0)\) and \((3,0)\), missing \((0,9)\).
- Option D: \((0,3)\) is not an intersection point (the parabola at \(x = 0\) is \(y = 9\), not \(3\)).

Answer:

C. \((-3,0)\), \((3,0)\), and \((0,9)\)