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Algebra / Systems of two linear equations in two variables Difficulty: Hard

$y = 6 x + 18$

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?

$- 6 x + y = 18$

$- 6 x + y = 22$

$- 12 x + y = 36$

$- 12 x + y = 18$

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Explanation

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 $\frac{D}{A} = \frac{E}{B}$ ; and the lines are distinct if the constants are not proportional, meaning $\frac{F}{C}$ is not equal to $\frac{D}{A}$ or $\frac{E}{B}$ . 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 $A = - 6$ , $upper B equals 1$ , and $upper C equals 18$ . The equation in choice B, $- 6 x + y = 22$ , is written in the form $D x + E y = F$ , where $D = - 6$ , $upper E equals 1$ , and $upper F equals 22$ . Therefore, $\frac{D}{A} = \frac{- 6}{- 6}$ , which can be rewritten as $\frac{D}{A} = 1$ ; $\frac{E}{B} = \frac{1}{1}$ , which can be rewritten as $\frac{E}{B} = 1$ ; and $\frac{F}{C} = \frac{22}{18}$ , which can be rewritten as $\frac{F}{C} = \frac{11}{9}$ . Since $\frac{D}{A} = 1$ , $\frac{E}{B} = 1$ , and $\frac{F}{C}$ 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.