OG详解-OG2 数学2 Q19

Choice B 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 parallel and distinct. Lines represented by equations in standard form, Ax +By =C and Dx +Ey =F, are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation, meaning DA​=EB​​ ; and the lines are distinct if the constants are not proportional, meaning FC​​ is not equal to DA​​ or EB​​ . The given equation, y =6x +18, can be written in standard form by subtracting 6x from both sides of the equation to yield -6xy + =18. Therefore, the given equation can be written in the form Ax +By =C, where A=-6, B =1, and C =18. The equation in choice B, -6x y + =22, is written in the form Dx +Ey =F, where D =-6, E =1, and F =22. Therefore,  DA​=−6−6​​ , which can be rewritten as DA​=1​ ;EB​=11​​ , which can be rewritten as EB​= 1; and FC​​= 2218​​, which can be rewritten as FC​​ = 119​​. SinceDA​​= 1, EB​​= 1 , and FC​​is not equal to 1, it follows that the given equation and the equation -6x y + =22 are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation -6x y + =22 has no solution. Thus, the equation in choice B could be the second equation in the system.
Choice A is incorrect. The equation -6xy + =18 and the given equation represent the same line in the xy-plane. Therefore, a system of these linear equations would have infinitely many solutions, rather than no solution. Choice C is incorrect. The equation -12xy + =36 and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution. Choice D is incorrect. The equation -12x y + =18 and the given equation represent lines in the xy-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.