OG详解-OG8 数学2 Q23

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, meaningDA​=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 -6x+y= 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 F​C​=2218​​ , which can be rewritten as FC​​  =  119​​ . Since  DA​​ = 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 -6x+y= 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 -12x+y= 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.