4 x - 9 y = 9 y + 5 h y = 2 + 4 x In the given

Question

4 x - 9 y = 9 y + 5 h y = 2 + 4 x In the given system of equations, h is a constant. If the system has no solution, what is the value of h ?

Accepted Answer

Choice D is correct. A system of two linear equations in two variables, x and y , has no solution if the lines represented by the equations in the xy -plane are distinct and parallel. The graphs of two lines in the xy -plane represented by equations in the form A x + B y = C , where A , B , and C are constants, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation. The first equation in the given system can be written in the form A x + B y = C by subtracting 9 y from both sides of the equation to yield 4 x - 18 y = 5 . The second equation in the given system can be written in the form A x + B y = C by subtracting 4 x from both sides of the equation to yield - 4 x + h y = 2 . The coefficient of x in this second equation, - 4 , is - 1 times the coefficient of x in the first equation, 4 . For the lines to be parallel, the coefficient of y in the second equation, h , must also be - 1 times the coefficient of y in the first equation, - 18 . Thus, h = - 1 ( - 18 ) , or h = 18 . Therefore, if the given system has no solution, the value of h is 18 . Choice A is incorrect. If the value of h is - 9 , then the given system would have one solution, rather than no solution. Choice B is incorrect. If the value of h is 0 , then the given system would have one solution, rather than no solution. Choice C is incorrect. If the value of h is 9 , then the given system would have one solution, rather than no solution.

Question 4-x-9-