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

$y equals, x squared, plus 3 x, minus 7, and, y minus 5 x, plus 8, equals 0$

How many solutions are there to the system of equations above?

There are exactly 4 solutions.

There are exactly 2 solutions.

There is exactly 1 solution.

There are no solutions.

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Explanation

Choice C is correct. The second equation of the system can be rewritten as $y equals, 5 x minus 8$ . Substituting $5 x minus 8$ for y in the first equation gives $5 x minus 8, equals, x squared, plus 3 x, minus 7$ . This equation can be solved as shown below:

$x squared, plus 3 x, minus 7, minus 5 x, plus 8, equals 0$

$x squared, minus 2 x, plus 1, equals 0$

$open parenthesis, x minus 1, close parenthesis, squared, equals 0$

$x equals 1$

Substituting 1 for x in the equation $y equals, 5 x minus 8$ gives $y equals negative 3$ . Therefore, $the ordered pair 1 comma negative 3$ is the only solution to the system of equations.

Choice A is incorrect. In the xy-plane, a parabola and a line can intersect at no more than two points. Since the graph of the first equation is a parabola and the graph of the second equation is a line, the system cannot have more than 2 solutions. Choice B is incorrect. There is a single ordered pair $x comma y$ that satisfies both equations of the system. Choice D is incorrect because the ordered pair $1 comma negative 3$ satisfies both equations of the system.