y = 18 y = - 3 x - 18 2 + 15 If the given equations are graphed in

Question

y = 18 y = - 3 x - 18 2 + 15 If the given equations are graphed in the xy -plane, at how many points do the graphs of the equations intersect?

Accepted Answer

Choice D is correct. A point x , y is a solution to a system of equations if it lies on the graphs of both equations in the xy -plane. In other words, a solution to a system of equations is a point x , y at which the graphs intersect. It&rsquo;s given that the first equation is y = 18 . Substituting 18 for y in the second equation yields 18 = - 3 x - 18 2 + 15 . Subtracting 15 from each side of this equation yields 3 = - 3 x - 18 2 . Dividing each side of this equation by - 3 yields - 1 = x - 18 2 . Since the square of a real number is at least 0 , this equation can't have any real solutions. Therefore, the graphs of the equations intersect at zero points. Alternate approach: The graph of the second equation is a parabola that opens downward and has a vertex at 18 , 15 . Therefore, the maximum value of this parabola occurs when y = 15 . The graph of the first equation is a horizontal line at 18 on the y -axis, or y = 18 . Since 18 is greater than 15 , or the horizontal line is above the vertex of the parabola, the graphs of these equations intersect at zero points. Choice A is incorrect. The graph of y = 15 , not y = 18 , and the graph of the second equation intersect at exactly one point. Choice B is incorrect. The graph of any horizontal line such that the value of y is less than 15 , not greater than 15 , and the graph of the second equation intersect at exactly two points. Choice C is incorrect and may result from conceptual or calculation errors.

Question y-18-y