2x + 3y = 7
10x + 15y = 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?

Question

2x + 3y = 7 10x + 15y = 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, − 3r ⁄ 2 + 7 ⁄ 2 and r, for x and y, respectively, in the first equation yields 2(− 3r ⁄ 2 + 7 ⁄ 2 ) + 3r = 7, which is equivalent to −3r + 7 + 3r = 7, or 7 = 7. Therefore, the point (− 3r ⁄ 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.

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