y = 6 x + 18 One of the equations in a system of two linear equations is given. The

Question

y = 6 x + 18 One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

Accepted Answer

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, A x + B y = C and D x + E y = F , are parallel if the coefficients for x and y in one equation are proportional to the corresponding coefficients in the other equation, meaning D A = E B ; and the lines are distinct if the constants are not proportional, meaning F C is not equal to D A or E B . The given equation, y = 6 x + 18 , can be written in standard form by subtracting 6 x from both sides of the equation to yield - 6 x + y = 18 . Therefore, the given equation can be written in the form A x + B y = C , where A = - 6 , B = 1 , and C = 18 . The equation in choice B, - 6 x + y = 22 , is written in the form D x + E y = F , where D = - 6 , E = 1 , and F = 22 . Therefore, D A = - 6 - 6 , which can be rewritten as D A = 1 ; E B = 1 1 , which can be rewritten as E B = 1 ; and F C = 22 18 , which can be rewritten as F C = 11 9 . Since D A = 1 , E B = 1 , and F C is not equal to 1 , it follows that the given equation and the equation - 6 x + y = 22 are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation - 6 x + 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 - 6 x + 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 - 12 x + 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 - 12 x + 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.

Question y-6-x-