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

$3 x = 36 y - 45$

One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?

$x equals 4 y$

$one third x equals 4 y$

$x = 12 y - 15$

$\frac{1}{3} x = 12 y - 15$

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Explanation

Choice B is correct. A system of two linear equations in two variables, $x$ and $y$ , has no solution when the lines in the xy-plane representing the equations are parallel and distinct. Two lines are parallel and distinct if their slopes are the same and their y-intercepts are different. The slope of the graph of the given equation, $3 x = 36 y - 45$ , in the xy-plane can be found by rewriting the equation in the form $y = m x + b$ , where $m$ is the slope of the graph and $left parenthesis 0 comma b right parenthesis$ is the y-intercept. Adding $45$ to each side of the given equation yields $3 x + 45 = 36 y$ . Dividing each side of this equation by $36$ yields $\frac{1}{12} x + \frac{5}{4} = y$ , or $y = \frac{1}{12} x + \frac{5}{4}$ . It follows that the slope of the graph of the given equation is $\frac{1}{12}$ and the y-intercept is $left parenthesis 0 comma five fourths right parenthesis$ . Therefore, the graph of the second equation in the system must also have a slope of $\frac{1}{12}$ , but must not have a y-intercept of $left parenthesis 0 comma five fourths right parenthesis$ . Multiplying each side of the equation given in choice B by $\frac{1}{4}$ yields $\frac{1}{12} x = y$ , or $y = \frac{1}{12} x$ . It follows that the graph representing the equation in choice B has a slope of $\frac{1}{12}$ and a y-intercept of $left parenthesis 0 comma 0 right parenthesis$ . Since the slopes of the graphs of the two equations are equal and the y-intercepts of the graphs of the two equations are different, the equation in choice B could be the second equation in the system.

Choice A is incorrect. This equation can be rewritten as $y = \frac{1}{4} x$ . It follows that the graph of this equation has a slope of $\frac{1}{4}$ , so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.

Choice C is incorrect. This equation can be rewritten as $y = \frac{1}{12} x + \frac{5}{4}$ . It follows that the graph of this equation has a slope of $\frac{1}{12}$ and a y-intercept of $left parenthesis 0 comma five fourths right parenthesis$ , so the system consisting of this equation and the given equation has infinitely many solutions, rather than no solution.

Choice D is incorrect. This equation can be rewritten as $y = \frac{1}{36} x + \frac{5}{4}$ . It follows that the graph of this equation has a slope of $\frac{1}{36}$ , so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.