5 x + 7 y = 1 a x + b y = 1 In the given pair of equations,

Question

5 x + 7 y = 1 a x + b y = 1 In the given pair of equations, a and b are constants. The graph of this pair of equations in the xy -plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?

Accepted Answer

Choice B is correct. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to - 1 divided by the slope of the second line. Each equation in the given pair of equations can be written in slope-intercept form, y = m x + b , where m is the slope of the graph of the equation in the xy -plane and 0 , b is the y -intercept. For the first equation, 5 x + 7 y = 1 , subtracting 5 x from both sides gives 7 y = - 5 x + 1 , and dividing both sides of this equation by 7 gives y = - 5 7 x + 1 7 . Therefore, the slope of the graph of this equation is - 5 7 . For the second equation, a x + b y = 1 , subtracting a x from both sides gives b y = - a x + 1 , and dividing both sides of this equation by b gives y = - a b x + 1 b . Therefore, the slope of the graph of this equation is - a b . Since the graph of the given pair of equations is a pair of perpendicular lines, the slope of the graph of the second equation, - a b , must be the negative reciprocal of the slope of the graph of the first equation, - 5 7 . The negative reciprocal of - 5 7 is - 1 - 5 7 , or 7 5 . Therefore, - a b = 7 5 , or a b = - 7 5 . Similarly, rewriting the equations in choice B in slope-intercept form yields y = - 10 7 x + 1 7 and y = - a 2 b x + 1 2 b . It follows that the slope of the graph of the first equation in choice B is - 10 7 and the slope of the graph of the second equation in choice B is - a 2 b . Since a b = - 7 5 , - a 2 b is equal to - 1 2 - 7 5 , or 7 10 . Since 7 10 is the negative reciprocal of - 10 7 , the pair of equations in choice B represents a pair of perpendicular lines. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

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