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Algebra / Systems of two linear equations in two variables Difficulty: Hard

$2 x + 3 y = 7$
$10 x + 15 y = 35$

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?

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Explanation

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, $- \frac{3 r}{2} + \frac{7}{2}$ and $r$ , for $x$ and $y$ , respectively, in the first equation yields $2 (- \frac{3 r}{2} + \frac{7}{2}) + 3 r = 7$ , which is equivalent to $- 3 r + 7 + 3 r = 7$ , or $7 equals 7$ . Therefore, the point $(- \frac{3 r}{2} + \frac{7}{2}, r)$ 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.