Set 3: Polynomial Functions (Advanced)

Choice C 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 Ax + By = 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 Ax + By = 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 Ax + By = C by subtracting 4 x from both sides of the equation to yield -4 x + hy = 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 h, or h = 18. Therefore, if the given system has no solution, the value of h is 18. Choice D 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 A is incorrect. If the value of h is 9, then the given system would have one solution, rather than no solution.