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

$y = (x - 2) (x + 4)$

$y = 6 x - 12$

Which ordered pair $(x, y)$ is the solution to the given system of equations?

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Explanation

Choice C is correct. The second equation in the given system of equations is $y = 6 x - 12$ . Substituting $6 x minus 12$ for $y$ in the first equation of the given system yields $6 x - 12 = (x - 2) (x + 4)$ . Factoring $6$ out of the left-hand side of this equation yields $6 (x - 2) = (x - 2) (x + 4)$ . An expression with a factor of the form $(x - a)$ is equal to zero when $x = a$ . Each side of this equation has a factor of $left parenthesis x minus 2 right parenthesis$ , so each side of the equation is equal to zero when $x equals 2$ . Substituting $2$ for $x$ into the equation $6 (x - 2) = (x - 2) (x + 4)$ yields $6 (2 - 2) = (2 - 2) (2 + 4)$ , or $0 equals 0$ , which is true. Substituting $2$ for $x$ into the second equation in the given system of equations yields $y = 6 (2) - 12$ , or $y equals 0$ . Therefore, the solution to the system of equations is the ordered pair $left parenthesis 2 comma 0 right parenthesis$ .

Choice A is incorrect and may result from switching the order of the solutions for $x$ and $y$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.