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

$- 15 x + 25 y = 65$

One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which of the following could be the second equation in the system?

$12 x + 20 y = 52$

$12 x + 20 y = - 52$

$- 12 x + 20 y = 52$

$- 12 x + 20 y = - 52$

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Explanation

Choice C is correct. It's given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Dividing both sides of the given equation by $5$ yields $- 3 x + 5 y = 13$ . Dividing both sides of choice C by $4$ also yields $- 3 x + 5 y = 13$ , so choice C is equivalent to the given equation. Thus, choice C could be the second equation in the system.

Choice A is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.

Choice B is incorrect. The system consisting of this equation and the given equation has one solution, not infinitely many solutions.

Choice D is incorrect. The system consisting of this equation and the given equation has no solution, not infinitely many solutions.