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

$8 x + 7 y = 9$
$24 x + 21 y = 27$

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 A is correct. Dividing both sides of the second equation in the given system by $3$ yields $8 x + 7 y = 9$ , which is the first equation in the given system. Therefore, the first and second equations represent the same line in the xy-plane. If the x- and y-coordinates of a point satisfy an equation, the point lies on the graph of the equation in the xy-plane. Choice A is a point with x-coordinate $r$ and y-coordinate $- \frac{8 r}{7} + \frac{9}{7}$ . Substituting $r$ for $x$ and $- \frac{8 r}{7} + \frac{9}{7}$ for $y$ in the equation $8 x + 7 y = 9$ yields $8 r + 7 (- \frac{8}{7} r + \frac{9}{7}) = 9$ . Applying the distributive property to the left-hand side of this equation yields $8 r - 8 r + 9 = 9$ . Combining like terms on the left-hand side of this equation yields $9 equals 9$ , so the coordinates of the point $(r, - \frac{8}{7} r + \frac{9}{7})$ satisfy both equations in the given system. Therefore, for each real number $r$ , the point $(r, - \frac{8}{7} r + \frac{9}{7})$ lies on the graph of each equation in the xy-plane for the given system.

Choice B 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.