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Geometry Difficulty 3.7 AMC 10/12 Find the answer United States

How many ordered pairs (x,y)(x, y) of real numbers satisfy the following system of equations?
x2+3y=9(x+y4)2=1 \begin{aligned} x^2 + 3y &= 9 \\ (|x| + |y| - 4)^2 &= 1 \end{aligned}

Pick one

Solution

Answer (D): The graph of the first equation is a parabola opening downward with vertex (0,3)(0, 3), passing through (3,0)(-3, 0) and (3,0)(3, 0). The second equation is satisfied if either x+y=5|x| + |y| = 5 or x+y=3|x| + |y| = 3. Therefore the graph of the second equation is a square with vertices (5,0)(5, 0), (0,5)(0, 5), (5,0)(-5, 0), and (0,5)(0, -5) together with a square with vertices (3,0)(3, 0), (0,3)(0, 3), (3,0)(-3, 0), and (0,3)(0, -3). As shown below, the two graphs intersect at 5 points—(0,3)(0, 3), (3,0)(3, 0), (3,0)(-3, 0), and 2 points on the larger square in the lower half-plane.

Figure 1

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.