- 12 x + 14 y = 36 - 6 x + 7 y = - 18 How many solutions

Question

- 12 x + 14 y = 36 - 6 x + 7 y = - 18 How many solutions does the given system of equations have?

Accepted Answer

Choice D is correct. A system of two linear equations in two variables, x and y , has zero solutions if the lines representing the equations in the xy -plane are distinct and parallel. Two lines are distinct and parallel if they have the same slope but different y -intercepts. Each equation in the given system can be written in slope-intercept form y = m x + b , where m is the slope of the line representing the equation in the xy -plane and 0 , b is the y -intercept. Adding 12 x to both sides of the first equation in the given system of equations, - 12 x + 14 y = 36 , yields 14 y = 12 x + 36 . Dividing both sides of this equation by 14 yields y = 6 7 x + 18 7 . It follows that the first equation in the given system of equations has a slope of 6 7 and a y -intercept of 0 , 18 7 . Adding 6 x to both sides of the second equation in the given system of equations, - 6 x + 7 y = - 18 , yields 7 y = 6 x - 18 . Dividing both sides of this equation by 7 yields y = 6 7 x - 18 7 . It follows that the second equation in the given system of equations has a slope of 6 7 and a y -intercept of 0 , - 18 7 . Since the slopes of these lines are the same and the y -intercepts are different, it follows that the given system of equations has zero solutions. Alternate approach: To solve the system by elimination, multiplying the second equation in the given system of equations, - 6 x + 7 y = - 18 , by - 2 yields 12 x - 14 y = 36 . Adding this equation to the first equation in the given system of equations, - 12 x + 14 y = 36 , yields ( - 12 x + 12 x ) + ( - 14 y + 14 y ) = 36 + 36 , or 0 = 72 . Since this equation isn't true, the given system of equations has zero solutions. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Question 12-x-1