2 x + 3 y = 7 10 x + 15 y = 35 For each real number r ,

Question

2 x + 3 y = 7 10 x + 15 y = 35 For each real number r , which of the following points lies on the graph of each equation in the xy -plane for the given system?

Accepted Answer

Choice B is correct. The two given equations are equivalent because the second equation can be obtained from the first equation by multiplying each side of the equation by 5 . Thus, the graphs of the equations are coincident, so if a point lies on the graph of one of the equations, it also lies on the graph of the other equation. A point x , y lies on the graph of an equation in the xy -plane if and only if this point represents a solution to the equation. It is sufficient, therefore, to find the point that represents a solution to the first given equation. Substituting the x - and y -coordinates of choice B, - 3 r 2 + 7 2 and r , for x and y , respectively, in the first equation yields 2 - 3 r 2 + 7 2 + 3 r = 7 , which is equivalent to - 3 r + 7 + 3 r = 7 , or 7 = 7 . Therefore, the point - 3 r 2 + 7 2 , r represents a solution to the first equation and thus lies on the graph of each equation in the xy -plane for the given system. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Question 2-x-3-