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Difficulty: Hard

$- 12 x + 14 y = 36$

$- 6 x + 7 y = - 18$

How many solutions does the given system of equations have?

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Explanation

Choice D is correct. A system of two linear equations in two variables, $x$ and $y$ , has zero solutions if the lines representing the equations in the xy-plane are distinct and parallel. Two lines are distinct and parallel if they have the same slope but different y-intercepts. Each equation in the given system can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope of the line representing the equation in the xy-plane and $left parenthesis 0 comma b right parenthesis$ is the y-intercept. Adding $12 x$ to both sides of the first equation in the given system of equations, $- 12 x + 14 y = 36$ , yields $14 y = 12 x + 36$ . Dividing both sides of this equation by $14$ yields $y = \frac{6}{7} x + \frac{18}{7}$ . It follows that the first equation in the given system of equations has a slope of $\frac{6}{7}$ and a y-intercept of $left parenthesis 0 comma StartFraction 18 Over 7 EndFraction right parenthesis$ . Adding $6 x$ to both sides of the second equation in the given system of equations, $- 6 x + 7 y = - 18$ , yields $7 y = 6 x - 18$ . Dividing both sides of this equation by $7$ yields $y = \frac{6}{7} x - \frac{18}{7}$ . It follows that the second equation in the given system of equations has a slope of $\frac{6}{7}$ and a y-intercept of $(0, - \frac{18}{7})$ . Since the slopes of these lines are the same and the y-intercepts are different, it follows that the given system of equations has zero solutions.

Alternate approach: To solve the system by elimination, multiplying the second equation in the given system of equations, $- 6 x + 7 y = - 18$ , by $negative 2$ yields $12 x - 14 y = 36$ . Adding this equation to the first equation in the given system of equations, $- 12 x + 14 y = 36$ , yields $(- 12 x + 12 x) + (- 14 y + 14 y) = 36 + 36$ , or $0 equals 72$ . Since this equation isn't true, the given system of equations has zero solutions.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.