3 x = 36 y - 45 One of the two equations in a system of linear equations is given.

Question

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?

Accepted Answer

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 0 , b 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 1 12 x + 5 4 = y , or y = 1 12 x + 5 4 . It follows that the slope of the graph of the given equation is 1 12 and the y -intercept is 0 , 5 4 . Therefore, the graph of the second equation in the system must also have a slope of 1 12 , but must not have a y -intercept of 0 , 5 4 . Multiplying each side of the equation given in choice B by 1 4 yields 1 12 x = y , or y = 1 12 x . It follows that the graph representing the equation in choice B has a slope of 1 12 and a y -intercept of 0 , 0 . 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 = 1 4 x . It follows that the graph of this equation has a slope of 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 = 1 12 x + 5 4 . It follows that the graph of this equation has a slope of 1 12 and a y -intercept of 0 , 5 4 , 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 = 1 36 x + 5 4 . It follows that the graph of this equation has a slope of 1 36 , so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.

Question 3-x-36