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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Medium

$Equation 1: x squared equals, 6 x plus y. Equation 2: y equals, negative 6 x, plus 36.$

A solution to the given system of equations is $the ordered pair x comma y$ . Which of the following is a possible value of xy ?

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Explanation

Choice A is correct. Solutions to the given system of equations are ordered pairs $x comma y$ that satisfy both equations in the system. Adding the left-hand and right-hand sides of the equations in the system yields $x squared, plus y, equals, 6 x plus negative 6 x, plus y, plus 36$ , or $x squared, plus y, equals, y plus 36$ . Subtracting y from both sides of this equation yields $x squared equals 36$ . Taking the square root of both sides of this equation yields $x equals 6$ and $x equals negative 6$ . Therefore, there are two solutions to this system of equations, one with an x-coordinate of 6 and the other with an x-coordinate of $negative 6$ . Substituting 6 for x in the second equation yields $y equals, negative 6 times 6, plus 36$ , or $y equals 0$ ; therefore, one solution is $the ordered pair 6 comma 0$ . Similarly, substituting $negative 6$ for x in the second equation yields $y equals, negative 6 times negative 6, plus 36$ , or $y equals 72$ ; therefore, the other solution is $the ordered pair negative 6 comma 72$ . It follows then that if $the ordered pair x comma y$ is a solution to the system, then possible values of $x y$ are $6 times 0, equals 0$ and $negative 6 times 72, equals negative 432$ . Only 0 is among the given choices.

Choice B is incorrect. This is the x-coordinate of one of the solutions, $the ordered pair 6 comma 0$ . Choice C is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This is the square of the x-coordinate of one of the solutions, $the ordered pair 6 comma 0$ .